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### Rsa number

RSA numbers are difficult to-factor composite numbers having exactly two prime factors (i.e., so-called semiprimes) that were listed in the Factoring Challenge of RSA Security®--a challenge that is now withdrawn and no longer active.While RSA numbers are much smaller than the largest known primes, their factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers and together, it can be extremely difficult to determine the factors if only their product is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.RSA Laboratories sponsored the RSA Factoring Challenge to encourage research into computational..

### Mrs. perkins's quilt

A Mrs. Perkins's quilt is a dissection of a square of side into a number of smaller squares. The name "Mrs. Perkins's Quilt" comes from a problem in one of Dudeney's books, where he gives a solution for . Unlike a perfect square dissection, however, the smaller squares need not be all different sizes. In addition, only prime dissections are considered so that patterns which can be dissected into lower-order squares are not permitted.The smallest numbers of squares needed to create relatively prime dissections of an quilt for , 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (OEIS A005670), the first few of which are illustrated above.On October 9-10, L. Gay (pers. comm. to E. Pegg, Jr.) discovered 18-square quilts for side lengths 88, 89, and 90, thus beating all previous records. The following table summarizes the smallest numbers of squares known to be needed for various side lengths , with those for (and possibly..

### Woodall prime

A Woodall prime is a Woodall numberthat is prime. The first few Woodall primes are 7, 23, 383, 32212254719, 2833419889721787128217599, ... (OEIS A050918), corresponding to , 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, ... (OEIS A002234). The following table summarizes large known Woodall primes. As of Mar. 2018, all have been checked (PrimeGrid).decimal digitsdate1467763441847Jun. 20072013992606279Aug. 20072367906712818Aug. 200737529481129757Dec. 2007170166025122515Mar. 2018

### Partition function p

, sometimes also denoted (Abramowitz and Stegun 1972, p. 825; Comtet 1974, p. 94; Hardy and Wright 1979, p. 273; Conway and Guy 1996, p. 94; Andrews 1998, p. 1), gives the number of ways of writing the integer as a sum of positive integers, where the order of addends is not considered significant. By convention, partitions are usually ordered from largest to smallest (Skiena 1990, p. 51). For example, since 4 can be written(1)(2)(3)(4)(5)it follows that . is sometimes called the number of unrestricted partitions, and is implemented in the Wolfram Language as PartitionsP[n].The values of for , 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (OEIS A000041). The values of for , 1, ... are given by 1, 42, 190569292, 24061467864032622473692149727991, ... (OEIS A070177).The first few prime values of are 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575), corresponding to indices 2, 3, 4, 5, 6, 13, 36, 77, 132,..

### Euler prime

Let a prime number generated by Euler's prime-generating polynomial be known as an Euler prime. Then the first few Euler primes occur for , 2, ..., 39, 42, 43, 45, ... (OEIS A056561), corresponding to the primes 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, ... (OEIS A005846).As of Feb. 2013, the largest known Euler prime is , which has 398204 decimal digits and was found by D. Broadhurst (https://primes.utm.edu/primes/page.php?id=111195).

### Wolstenholme prime

A prime is called a Wolstenholme prime if the central binomial coefficient(1)or equivalently if(2)where is the th Bernoulli number and the congruence is fractional.A prime is a Wolstenholme prime if and only if(3)where the congruence is again fractional.The only known Wolstenholme primes are 16843 and 2124679 (OEIS A088164). There are no others up to (McIntosh 2004).

### Palindromic prime

A palindromic prime is a number that is simultaneously palindromic and prime. The first few (base-10) palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, ... (OEIS A002385; Beiler 1964, p. 228). The number of palindromic primes less than a given number are illustrated in the plot above. The number of palindromic numbers having , 2, 3, ... digits are 4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, ... (OEIS A016115; De Geest) and the total number of palindromic primes less than 10, , , ... are 4, 5, 20, 20, 113, 113, 781, ... (OEIS A050251). Gupta (2009) has computed the numbers of palindromic primes up to .The following table lists palindromic primes in various small bases. OEISbase- palindromic primes2A11769711, 101, 111, 10001, 11111, 1001001, 1101011, ...3A1176982, 111, 212, 12121, 20102, 22122, ...4A1176992, 3, 11, 101, 131, 323, 10001, 11311, 12121, ...5A1177002, 3, 111, 131, 232, 313, 414, 10301, 12121,..

### Euler number

The Euler numbers, also called the secant numbers or zig numbers, are defined for by(1)(2)where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.A different sort of Euler number, the Euler number of a finite complex , is defined by(3)This Euler number is a topological invariant.To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes.Some values of the (secant) Euler numbers are(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(OEIS A000364).The slightly different convention defined by(16)(17)is..

### Wilson prime

A Wilson prime is a prime satisfyingwhere is the Wilson quotient, or equivalently,The first few Wilson primes are 5, 13, and 563 (OEIS A007540). Crandall et al. (1997) showed there are no others less than (McIntosh 2004), a limit that has subsequently been increased to (Costa et al. 2012).

### Wieferich prime

A Wieferich prime is a prime which is a solution to the congruence equation(1)Note the similarity of this expression to the special case of Fermat'slittle theorem(2)which holds for all odd primes. The first few Wieferich primes are 1093, 3511, ... (OEIS A001220), with none other less than (Lehmer 1981, Crandall 1986, Crandall et al. 1997), a limit since increased to (McIntosh 2004) and subsequently to by PrimeGrid as of November 2015.Interestingly, one less than these numbers have suggestive periodic binaryrepresentations(3)(4)(Johnson 1977).If the first case of Fermat's last theorem is false for exponent , then must be a Wieferich prime (Wieferich 1909). If with and relatively prime, then is a Wieferich prime iff also divides . The conjecture that there are no three consecutive powerful numbers implies that there are infinitely many non-Wieferich primes (Granville 1986; Ribenboim 1996, p. 341; Vardi 1991). In addition, the abc..

### Odd perfect number

In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why an odd perfect number could not exist (Dickson 2005, p. 12). Descartes was therefore among the first to consider the existence of odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers (Dickson 2005, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect number could not exist. Like Frenicle, Euler also considered odd perfect numbers.To this day, it is not known if any odd perfect numbers exist, although..

### Weird number

A "weird number" is a number that is abundant (i.e., the sum of proper divisors is greater than the number) without being pseudoperfect (i.e., no subset of the proper divisors sums to the number itself). The pseudoperfect part of the definition means that finding weird numbers is a case of the subset sum problem.Since prime numbers are deficient, prime numbers are not weird. Similarly, since multiples of 6 are pseudoperfect, no weird number is a multiple of 6.The smallest weird number is 70, which has proper divisors 1, 2, 5, 7, 10, 14, and 35. These sum to 74, which is greater that the number itself, so 70 is abundant, and no subset of them sums to 70. In contrast, the smallest abundant number is 12, which has proper divisors 1, 2, 3, 4, and 6. These sum to 16, so 12 is abundant, but the subset sum equals 12, so 12 is not weird.The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ...(OEIS A006037).An infinite number of weird..

### Number field sieve

An extremely fast factorization method developed by Pollard which was used to factor the RSA-130 number. This method is the most powerful known for factoring general numbers, and has complexity(1)reducing the exponent over the continued fraction factorization algorithm and quadratic sieve. There are three values of relevant to different flavors of the method (Pomerance 1996). For the "special" case of the algorithm applied to numbers near a large power,(2)for the "general" case applicable to any odd positive number which is not a power,(3)and for a version using many polynomials (Coppersmith1993),(4)

### Euclid number

Euclid's second theorem states that the number of primes is infinite. The proof of this can be accomplished using the numbers(1)(2)known as Euclid numbers, where is the th prime and is the primorial.The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (OEIS A006862; Tietze 1965, p. 19).The indices of the first few prime Euclid numbers are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, ... (OEIS A014545), so the first few Euclid primes (commonly known as primorial primes) are 3, 7, 31, 211, 2311, 200560490131, ... (OEIS A018239). The largest known Euclid number is , and it is not known if there are an infinite number of prime Euclid numbers (Guy 1994, Ribenboim 1996).The largest factors of for , 2, ... are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (OEIS A002585)...

### Elliptic curve primality proving

Elliptic curve primality proving, abbreviated ECPP, is class of algorithms that provide certificates of primality using sophisticated results from the theory of elliptic curves. A detailed description and list of references are given by Atkin and Morain (1990, 1993).Adleman and Huang (1987) designed an independent algorithm using hyperellipticcurves of genus two.ECPP is the fastest known general-purpose primality testing algorithm. ECPP has a running time of . As of 2004, the program PRIMO can certify a 4769-digit prime in approximately 2000 hours of computation (or nearly three months of uninterrupted computation) on a 1 GHz processor using this technique. As of 2009, the largest prime certified using this technique was the 11th Mills' prime (https://primes.utm.edu/primes/page.php?id=77907)which has decimal digits. The proof was performed using a distributed computation that started in September 2005 and ended in June 2006..

### Wagstaff prime

A Wagstaff prime is a prime number of the form for a prime number. The first few are given by , 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, and 4031399 (OEIS A000978), with and larger corresponding to probable primes. These values correspond to the primes with indices , 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, ... (OEIS A123176).The Wagstaff primes are featured in the newMersenne prime conjecture.There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving.A Wagstaff prime can also be interpreted as a repunit prime of base , asif is odd, as it must be for the above number to be prime.Some of the largest known Wagstaff probable primes are summarized in the following..

### Natural logarithm of 10 digits

The numerical value of is given by(OEIS A002392). It was computed to decimal digits by S. Kondo on May 20, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082, ... (OEIS A228243).-constant primes occur at 1, 2, 40, 242, 842, 1541, 75067, ... decimal digits (OEIS A228240).The starting positions of the first occurrence of , 1, ... in the decimal expansion of (including the initial 2 and counting it as the first digit) are 3, 21, 1, 2, 13, 5, 17, 22, ... (OEIS A229197).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 22, 701, 7486, 88092, 1189434, 13426407, ... (OEIS A229124), which end at digits 7, 38, 351, 8493, 33058, 362945, ... (OEIS A229126).The digit strings 0123456789 first occurs starting at position 3349545080, but 9876543210 does not occur in the first..

### Elliptic curve factorization method

The elliptic curve factorization method, abbreviated ECM and sometimes also called the Lenstra elliptic curve method, is a factorization algorithm that computes a large multiple of a point on a random elliptic curve modulo the number to be factored . It tends to be faster than the Pollard rho factorization and Pollard p-1 factorization methods.Zimmermann maintains a table of the largest factors found using the ECM. As of Jan. 2009, the largest prime factor found using the ECM had 67 decimal digits. This factor of was found by B. Dodson on Aug. 24, 2006 (Zimmermann).

### Unitary amicable pair

A pair of numbers and such thatwhere is the unitary divisor function. Hagis (1971) and García (1987) give 82 such pairs. The first few are (114, 126), (1140, 1260), (18018, 22302), (32130, 40446), ... (OEIS A002952 and A002953; Pedersen).On Jan. 30, 2004, Y. Kohmoto discovered the largest known unitary amicable pair, where each member has 317 digits.Kohmoto calls a unitary amicable pair whose members are squareful a proper unitary amicable pair.

### Earls sequence

The Earls sequence gives the starting position in the decimal digits of (or in general, any constant), not counting digits to the left of the decimal point, at which a string of copies of the number first occurs. The following table gives generalized Earls sequences for various constants, including .constantOEISsequenceApéry's constantA22907410, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ...Catalan's constantA2248192, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ...Champernowne constantA2248961, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ...Copeland-Erdős constantA2248975, 113, 1181, 21670, 263423, 7815547, 35619942, 402720247, 450680638eA2248282, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011Euler-Mascheroni constantA2248265, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186Glaisher-Kinkelin constantA2257637,..

### Natural logarithm of 2 digits

The decimal expansion of the natural logarithmof 2 is given by(OEIS A002162). It was computed to decimal digits by S. Kondo on May 14, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693, 2539803904 (OEIS A228242).-constant primes occur at 321, 466, 1271, 15690, 18872, 89973, ... decimal digits (OEIS A228226).The starting positions of the first occurrence of , 1, ... in the decimal expansion of are 9, 4, 22, 3, 5, 10, 1, 6, 8, ... (OEIS A100077).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 2, 98, 604, 1155, 46847, 175403, ... (OEIS A036901), which end at digits 22, 444, 7655, 98370, 1107795, 12983306, ... (OEIS A036905).The digit string 0123456789 occurs starting at positions 3157027485, 8102152328, ... in the decimal digits of , and 9876543210 occurs starting..

### Twin primes

Twin primes are pairs of primes of the form (, ). The term "twin prime" was coined by Paul Stäckel (1862-1919; Tietze 1965, p. 19). The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (OEIS A014574). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (OEIS A001359 and A006512).All twin primes except (3, 5) are of the form .It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem, which states that the number obtained by adding the reciprocals of the odd twin primes,(1)converges to a definite number ("Brun's constant"), which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1996, p. 201)...

### E digits

The constant e with decimal expansion(OEIS A001113) can be computed to digits of precision in 10 CPU-minutes on modern hardware. was computed to digits by P. Demichel, and the first have been verified by X. Gourdon on Nov. 21, 1999 (Plouffe). was computed to decimal digits by S. Kondo on Jul. 5, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011, ... (OEIS A224828).The starting positions of the first occurrence of in the decimal expansion of (including the initial 2 and counting it as the first digit) are 14, 3, 1, 18, 11, 12, 21, 2, ... (OEIS A088576).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 6, 12, 548, 1769, 92994, ... (OEIS A036900), which end at digits 21, 372, 8092, 102128, ... (OEIS A036904).The digit sequence 0123456789..

### Multiperfect number

A number is -multiperfect (also called a -multiply perfect number or -pluperfect number) iffor some integer , where is the divisor function. The value of is called the class. The special case corresponds to perfect numbers , which are intimately connected with Mersenne primes (OEIS A000396). The number 120 was long known to be 3-multiply perfect () sinceThe following table gives the first few for , 3, ..., 6.2A0003966, 28, 496, 8128, ...3A005820120, 672, 523776, 459818240, 1476304896, 510011801604A02768730240, 32760, 2178540, 23569920, ...5A04606014182439040, 31998395520, 518666803200, ...6A046061154345556085770649600, 9186050031556349952000, ...Lehmer (1900-1901) proved that has at least three distinct prime factors, has at least four, at least six, at least nine, and at least 14, etc.As of 1911, 251 pluperfect numbers were known (Carmichael and Mason 1911). As of 1929, 334 pluperfect numbers were known, many of them found..

### Titanic prime

In the 1980s, Samuel Yates defined a titanic prime to be a prime number of at least 1000 decimal digits. The smallest titanic prime is . As of 1990, more than 1400 were known (Ribenboim 1990). By 1995, more than were known, and many tens of thousands are known today. The largest prime number known as of December 2018 is the Mersenne prime , which has a whopping decimal digits.

### Double mersenne number

A double Mersenne number is a number of the formwhere is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, 9223372036854775807, ... (OEIS A077585).A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne prime can be prime only for prime , a double Mersenne prime can be prime only for prime , i.e., a Mersenne prime. Double Mersenne numbers are prime for , 3, 5, 7, corresponding to the sequence 7, 127, 2147483647, 170141183460469231731687303715884105727, ... (OEIS A077586).The next four , , , and have known factors summarized in the following table. The status of all other double Mersenne numbers is unknown, with being the smallest unresolved case. Since this number has 694127911065419642 digits, it is much too large for the usual Lucas-Lehmer test to be practical. The only possible method of determining the status of this number is therefore attempting to find small divisors..

### Theodorus's constant digits

Theodorus's constant has decimal expansion(OEIS A002194). It was computed to decimal digits by E. Weisstein on Jul. 23, 2013.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874).-constant primes occur at 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 5, 1, 4, 3, 23, 6, 12, 2, 8, 18, ... (OEIS A229200).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 4, 91, 184, 5566, 86134, 35343, ... (OEIS A000000), which end at digits 23, 378, 7862, 77437, 1237533, 16362668, ... (OEIS A000000).The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does,..

### Mertens conjecture

Given the Mertens function defined by(1)where is the Möbius function, Stieltjes claimed in an 1885 letter to Hermite that stays within two fixed bounds, which he suggested could probably be taken to be (Havil 2003, p. 208). In the same year, Stieltjes (1885) claimed that he had a proof of the general result. However, it seems likely that Stieltjes was mistaken in this claim (Derbyshire 2004, pp. 160-161). Mertens (1897) subsequently published a paper opining based on a calculation of that Stieltjes' claim(2)for was "very probable."The Mertens conjecture has important implications, since the truth of any equalityof the form(3)for any fixed (the form of the Mertens conjecture with ) would imply the Riemann hypothesis. In fact, the statement(4)for any is equivalent to the Riemann hypothesis (Derbyshire 2004, p. 251).Mertens (1897) verified the conjecture for , and this was subsequently extended to by..

### Th&acirc;bit ibn kurrah prime

A Thâbit ibn Kurrah prime, sometimes called a 321-prime, is a Thâbit ibn Kurrah number (i.e., a number of the form for nonnegative integer ) that is prime.The indices for the first few Thâbit ibn Kurrah primes are 0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, ... (OEIS A002235), corresponding to the primes 2, 5, 11, 23, 47, 191, 383, 6143, ... (OEIS A007505). Riesel (1969) extended the search to . A search for larger primes was coordinated by P. Underwood. PrimeGrid has continued that search and has checked values of up to as of Nov. 2015 (PrimeGrid). The table below summarizes the largest known Thâbit ibn Kurrah primes.digitsdiscovererPrimeGrid (Dec. 2005; https://primes.utm.edu/primes/page.php?id=76506)PrimeGrid (Mar. 2007; https://primes.utm.edu/primes/page.php?id=79671)PrimeGrid (Apr. 2008; https://primes.utm.edu/primes/page.php?id=84769)PrimeGrid..

### Mersenne prime

A Mersenne prime is a Mersenne number, i.e., anumber of the formthat is prime. In order for to be prime, must itself be prime. This is true since for composite with factors and , . Therefore, can be written as , which is a binomial number that always has a factor .The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (OEIS A000668) corresponding to indices , 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (OEIS A000043).Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number. L. Welsh maintains an extensive bibliography and history of Mersenne numbers.It has been conjectured that there exist an infinite number of Mersenne primes. Fitting a line through the origin to the asymptotic number of Mersenne primes with for the first 51 (known) Mersenne primes gives a best-fit line with , illustrated above. If the line is not restricted to pass through..

### Curve of constant width

Curves which, when rotated in a square, make contact with all four sides. Such curves are sometimes also known as rollers.The "width" of a closed convex curve is defined as the distance between parallel lines bounding it ("supporting lines"). Every curve of constant width is convex. Curves of constant width have the same "width" regardless of their orientation between the parallel lines. In fact, they also share the same perimeter (Barbier's theorem). Examples include the circle (with largest area), and Reuleaux triangle (with smallest area) but there are an infinite number. A curve of constant width can be used in a special drill chuck to cut square "holes."A generalization gives solids of constant width. These do not have the same surface area for a given width, but their shadows are curves of constant width with the same width!..

### Ternary

The base-3 method of counting in which only the digits 0, 1, and 2 are used. Ternary numbers arise in a number of problems in mathematics, including some problems of weighing. However, according to Knuth (1998), "no substantial application of balanced ternary notation has been made" (balanced ternary uses digits , 0, and 1 instead of 0, 1, and 2).The illustration above shows a graphical representation of the numbers 0 to 25 in ternary, and the following table gives the ternary equivalents of the first few decimal numbers. The concatenation of the ternary digits of the consecutive numbers 0, 1, 2, 3, ... gives (0), (1), (2), (1, 0), (1, 1), (1, 2), (2, 0), ... (OEIS A054635).111110221210221211022211310131112321241114112242205121512025221620161212622272117122271000822182002810019100192012910021010120202301010Ternary digits have the following multiplicationtable.0120000101220211A ternary representation can..

### Lucas prime

The first few prime Lucas numbers are 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, ... (OEIS A005479), corresponding to indices , 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849, ... (Dubner and Keller 1999, Lifchitz and Lifchitz; OEIS A001606). Only those up to index 56003 have been proven prime (Broadhurst and Irvine 2006; https://primes.utm.edu/primes/page.php?id=77992). As of Apr. 2009, the largest known Lucas probable prime is , which has decimal digits (R. Lifchitz, Mar. 2009)...

### Cunningham number

A Cunningham number is a binomial number of the form with and positive integers. Bases which are themselves powers need not be considered since they correspond to . Prime numbers of the form are very rare.A necessary (but not sufficient) condition for to be prime is that be of the form . Numbers of the form are called Fermat numbers, and the only known primes occur for , , , , and (i.e., , 1, 2, 3, 4). The only other primes for nontrivial and are , , and . is always divisible by 3 when is odd.Primes of the form are also very rare. The Mersenne numbers are known to be prime only for 44 values, the first few of which are , 3, 5, 7, 13, 17, 19, ... (OEIS A000043). Such numbers are known as Mersenne primes. There are no other primes for nontrivial and .In 1925, Cunningham and Woodall (1925) gathered together all that was known about the primality and factorization of the numbers and published a small book of tables. These tables collected from scattered sources the known prime..

### Tau function prime

is prime for , 458329, 942841, 966289, 1510441, ... (OEIS A135430). These values are also known as Lehmer-Ramanujan numbers or LR numbers since the first of them was found by Lehmer (1965). The corresponding primes have explicit values given by , , ... (OEIS A265913).It is known that if is prime, then must be an odd square.Large values of for which is a (probable) prime are summarized in the table below (Lifchitz and Lifchitz).decimal digitsdiscoverer180524N. Lygeros and O. Rozier (May 2015)258571N. Lygeros and O. Rozier (May 2015)282048N. Lygeros and O. Rozier (May 2015)498503N. Lygeros and O. Rozier (May 2015)555339N. Lygeros and O. Rozier (Sep. 2015)

### Cunningham chain

A sequence of primes is a Cunningham chain of the first kind (second kind) of length if () for , ..., . Cunningham primes of the first kind are Sophie Germain primes.It is conjectured there are arbitrarily long Cunningham chains. The longest known Cunningham chains are of length 17, with the first examples found corresponding to (first kind; J. Wroblewski, May 2008) and (second kind; J. Wroblewski, Jun. 2008).The smallest prime beginning a complete Cunningham chain of the first kind of lengths , 2, ... are 13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769, 26089808579, ... (OEIS A005602).The smallest prime beginning a complete Cunningham chain of the second kind of lengths , 2, ... are 11, 7, 2, 2131, 1531, 33301, 16651, 15514861, 857095381, 205528443121, ... (OEIS A005603)...

### Strong pseudoprime

A strong pseudoprime to a base is an odd composite number with (for odd) for which either(1)or(2)for some , 1, ..., (Riesel 1994, p. 91). Note that Guy (1994, p. 27) restricts the definition of strong pseudoprimes to only those satisfying (1).The definition is motivated by the fact that a Fermat pseudoprime to the base satisfies(3)But since is odd, it can be written , and(4)If is prime, it must divide at least one of the factors, but can't divide both because it would then divide their difference(5)Therefore,(6)so write to obtain(7)If divides exactly one of these factors but is composite, it is a strong pseudoprime. A composite number is a strong pseudoprime to at most 1/4 of all bases less than itself (Monier 1980, Rabin 1980). The strong pseudoprimes provide the basis for Miller's primality test and Rabin-Miller strong pseudoprime test.A strong pseudoprime to the base is also an Euler pseudoprime to the base (Pomerance et al. 1980)...

### Lehmer's mahler measure problem

An unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum Mahler measure for a univariate polynomial that is not a product of cyclotomic polynomials. Lehmer (1933) conjectured that if is such an integer polynomial, then(1)(2)where , denoted by Lehmer (1933) and by Hironaka (2009), is the largest positive root of this polynomial. The roots of this polynomial, plotted in the left figure above, are very special, since 8 of the 10 lie on the unit circle in the complex plane. The roots of the polynomials (represented by half their coefficients) giving the two next smallest known Mahler measures are also illustrated above (Mossinghoff 1998, p. S11).The best current bound is that of Smyth (1971), who showed that , where is a nonzero nonreciprocal polynomial that is not a product of cyclotomic polynomials (Everest 1999), and is the real root of . Generalizations of Smyth's result have been constructed by Lloyd-Smith..

### Cubic number

A cubic number is a figurate number of the form with a positive integer. The first few are 1, 8, 27, 64, 125, 216, 343, ... (OEIS A000578). The protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time recites the cubic numbers to calm himself and prevent himself from wanting to hit someone (Haddon 2003, p. 213).The generating function giving the cubic numbersis(1)The hex pyramidal numbers are equivalent tothe cubic numbers (Conway and Guy 1996).The plots above show the first 255 (top figure) and 511 (bottom figure) cubic numbers represented in binary.Pollock (1843-1850) conjectured that every number is the sum of at most 9 cubic numbers (Dickson 2005, p. 23). As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (, proved by Dickson, Pillai, and Niven in the early twentieth century), that every "sufficiently large" integer..

### Latin square

An Latin square is a Latin rectangle with . Specifically, a Latin square consists of sets of the numbers 1 to arranged in such a way that no orthogonal (row or column) contains the same number twice. For example, the two Latin squares of order two are given by(1)the 12 Latin squares of order three are given by(2)and two of the whopping 576 Latin squares of order 4 are given by(3)The numbers of Latin squares of order , 2, ... are 1, 2, 12, 576, 161280, ... (OEIS A002860). The number of isotopically distinct Latin squares of order , 2, ... are 1, 1, 1, 2, 2, 22, 564, 1676267, ... (OEIS A040082).A pair of Latin squares is said to be orthogonal if the pairs formed by juxtaposing the two arrays are all distinct. For example, the two Latin squares(4)are orthogonal. The number of pairs of orthogonal Latin squares of order , 2, ... are 0, 0, 36, 3456, ... (OEIS A072377).The number of Latin squares of order with first row given by is the same as the number of fixed diagonal Latin..

### Cubic nonplanar graph

A cubic nonplanar graph is a graph that is both cubicand nonplanar.The following table summarizes some named nonplanar cubic graphs.graph utility graph6Petersen graph10Franklin graph12Heawood graph14Möbius-Kantor graph16first Blanusa snark18second Blanusa snark18Pappus graph18Desargues graph20flower snark20McGee graph24Coxeter graph28double star snark30Levi graph30Dyck graph32Szekeres snark50Gray graph54Balaban 10-cage70Foster graph90Biggs-Smith graph102Balaban 11-cage112Tutte 12-cage126The largest cubic nonplanar graphs having diameters 3 and 4 are illustrated above. They have 20 and 38 vertices, respectively.

### Square number

A square number, also called a perfect square, is a figurate number of the form , where is an integer. The square numbers for , 1, ... are 0, 1, 4, 9, 16, 25, 36, 49, ... (OEIS A000290).A plot of the first few square numbers represented as a sequence of binary bits is shown above. The top portion shows to , and the bottom shows the next 510 values.The generating function giving the squarenumbers is(1)The st square number is given in terms of the th square number by(2)since(3)which is equivalent to adding a gnomon to the previoussquare, as illustrated above.The th square number is equal to the sum of the st and th triangular numbers,(4)(5)as can seen in the above diagram, in which the st triangular number is represented by the white triangles, the th triangular number is represented by the black triangles, and the total number of triangles is the square number (R. Sobel, pers. comm.).Square numbers can also be generated by taking the product of two consecutive..

### Large prime

The largest known prime numbers are Mersenne primes, the largest of these known as of September 2013 bing , which has a whopping decimal digits.As of Sep. 2013, the largest known probable primes are the Wagstaff primes and , both found by R. Propper in Sep. 2013 and which have and decimal digits, respectively. Other large known probable primes are the "dual Sierpinski numbers" (Moore 2009) given by and , which have and decimal digits, respectively.A prime with at least 1000 decimal digits is (or used to be) known as a titanicprime.

### Cube

The cube is the Platonic solid (also called the regular hexahedron). It is composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol .The cube is illustrated above, together with a wireframe version and a net(top figures). The bottom figures show three symmetric projections of the cube.The cube is implemented in the Wolfram Language as Cube[], and precomputed properties are available as PolyhedronData["Cube"].There are a total of 11 distinct nets for the cube (Turney 1984-85, Buekenhout and Parker 1998, Malkevitch), illustrated above, the same number as the octahedron. Questions of polyhedron coloring of the cube can be addressed using the Pólya enumeration theorem.A cube with unit edge lengths is called a unit cube.The surface area and volume of a cube with edge..

### Sophie germain prime

A prime is said to be a Sophie Germain prime if both and are prime. The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ... (OEIS A005384). It is not known if there are an infinite number of Sophie Germain primes (Hoffman 1998, p. 190).The numbers of Sophie Germain primes less than for , 2, ... are 3, 10, 37, 190, 1171, 7746, 56032, ... (OEIS A092816).The largest known proven Sophie Germain prime pair as of Feb. 29, 2016 is given by where(Caldwell), each of which has decimal digits (PrimeGrid).The definition of Sophie Germain primes and the value of the largest then-known suchprime were mentioned by the characters Hal and Catherine in the 2005 film Proof.Sophie Germain primes of the form correspond to the indices of composite Mersenne numbers .Around 1825, Sophie Germain proved that the first case of Fermat's last theorem is true for such primes, i.e., if is a Sophie Germain prime, then there do not exist integers..

### Cousin primes

Pairs of primes of the form (, ) are called cousin primes. The first few are (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), ... (OEIS A023200 and A046132).A large pair of cousin (proven) primes start with(1)where is a primorial. These primes have 10154 digits and were found by T. Alm, M. Fleuren, and J. K. Andersen (Andersen 2005).As of Jan. 2006, the largest known pair of cousin (probable) primes are(2)which have 11311 digits and were found by D. Johnson in May 2004.According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes,(3)(4)where (OEIS A114907) is the twin primes constant.An analogy to Brun's constant, the constant(5)(omitting the initial term ) can be defined. Using cousin primes up to , the value of is estimated as(6)..

### Soldner's constant digits

Ramanujan calculated (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is(OEIS A070769; Derbyshire 2004, p. 114). The first decimal digits were computed by E. Weisstein on Oct. 7, 2013.-constant primes occur for 4, 144, 227, 444, 19474, ... (OEIS A122422) decimal digits.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 3, 42, 178, 10013, 31567, 600035, 1253449, ... (OEIS A229071).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 17, 1, 8, 5, 2, 3, 6, 34, 11, ... (OEIS A229201).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 465, 102, 5858, 48441, ... (OEIS A000000), which end at digits 34, 512, 7454, 92508, 1414058, ... (OEIS A000000).The digit sequences 0123456789 and 9876543210 do not occur..

### Khinchin's constant digits

The numerical value of Khinchin's constant is given by(OEIS A002210). However, the numerical value of is notoriously difficult to calculate to high precision. Bailey et al. (1997) computed to 7350 digits, and the current record is digits, computed by Xavier Gourdon in 1997 with a computation requiring 22 hours and 23 minutes (Plouffe).The Earls sequence (starting position of copies of the digit ) for Khinchin's constant is given for , 2, ... by 9, 42, 1799, 494, 5760, ... (OEIS A224836), with the term being larger than .-constant primes occur at 1, 407, 878, 4443, 4981, 6551, 13386, 28433, ... decimal digits (OEIS A118327).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 2 and counting it as the first digit) are 8, 10, 1, 14, 5, 4, 2, 23, 3, 22, ... (OEIS A229196).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 43,..

### Smith number

A Smith number is a composite number the sum of whose digits is the sum of the digits of its prime factors (excluding 1). (The primes are excluded since they trivially satisfy this condition). One example of a Smith number is the beast number(1)since(2)Another Smith number is(3)since(4)The first few Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, ... (OEIS A006753). The corresponding digits sums are 4, 4, 9, 13, 13, 13, 4, 13, 4, 13, 13, 13, 13, ... (OEIS A050218). McDaniel (1987a) showed that there are an infinite number of Smith numbers.A generalized -Smith number can also be defined as a number satisfying , where is the sum of the digits of 's prime factors and is the usual sum of 's digits. The following table gives the first few -Smith numbers for small integers and their inverses.OEIS-Smith numbersA0502256969, 19998, 36399, 39693, 66099, 69663, ...A05022488, 169, 286, 484, 598, 682, 808, 844, 897, ...1A0067534, 22, 27,..

### Constant primes

Let be a prime with digits and let be a constant. Call an "-prime" if the concatenation of the first digits of (ignoring the decimal point if one is present) give . Constant primes are therefore a special type of integer sequence primes, with e-primes, pi-primes, and phi-primes being perhaps the most prominent examples.The following table summarizes the indices of known constant primes for some named mathematical constants.constantname of primesOEIS giving primeApéry's constantA11933410, 55, 109, 141Catalan's constantA11832852, 276, 25477Champernowne constantA07162010, 14, 24, 235, 2804, 4347, 37735, 68433Copeland-Erdős constantA2275301, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048ee-primeA0641181, 3, 7, 85, 1781, 2780, 112280, 155025Euler-Mascheroni constantA0658151, 3, 40, 185, 1038, 22610, 179849Glaisher-Kinkelin constantA1184207, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692Golomb-Dickman..

### Keith number

A Keith number is an -digit integer such that if a Fibonacci-like sequence (in which each term in the sequence is the sum of the previous terms) is formed with the first terms taken as the decimal digits of the number , then itself occurs as a term in the sequence. For example, 197 is a Keith number since it generates the sequence 1, 9, 7, , , , , , ... (Keith). Keith numbers are also called repfigit (repetitive fibonacci-like digit) numbers.There is no known general technique for finding Keith numbers except by exhaustive search. Keith numbers are much rarer than the primes, with only 84 Keith numbers with digits. The first few are 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, ... (OEIS A007629). As of Mar. 31, 2006, there are 95 known Keith numbers (Keith). The number of Keith numbers having , 2, ... digits are 0, 6, 2, 9, 7, 10, 2, 3, 2, 0, 2, 4, 2, 3, 3, 3, 5, 3, 5, 3, 1, 1, 3, 1, 1, 3, 7, 1, 2, 5, 2, 4, 6, 3, ... (OEIS A050235),..

### Constant digit scanning

Scan the decimal expansion of a constant (including any digits to the left of the decimal point) until all -digit strings have been seen (including 0-padded strings). The following table then gives the number of digits that must be scanned to encounter all , 2, ...-digit strings (where "number of digits" means the ending-not starting-digit of an -digit string) together with the last -digit string encountered.constantOEISsequenceApéry's constantA03690623, 457, 7839, 83054, 1256587, 13881136, 166670757, ...A0369027, 89, 211, 2861, 43983, 29270, 8261623, ...Catalan's constantA00000032, 716, 7700, 86482, 1143572, ...A0000008, 45, 529, 2679, 24200, ...Champernowne constantA07229011, 192, 2893, 38894, 488895, 5888896, 68888897, 788888898, 8888888899, ...Copeland-Erdős constantA00000048, 934, 24437, 366399, 4910479, 49672582, ...A0000000, 84, 504, 8580, 07010, 088880, ...eA03690421, 372, 8092,..

### Constant

A constant, sometimes also called a "mathematical constant," is any well-defined real number which is significantly interesting in some way. In this work, the term "constant" is generally reserved for real nonintegral numbers of interest, while "number" is used to refer to interesting integers (e.g., Brun's constant, but beast number). However, in contexts such as linear combination, the term "constant" is generally used to mean "scalar" or "real number," and need not exclude integer values.A function, equation, etc., is said to "be constant" (or be a constant function) if it always assumes the same value independent of how its parameters are varied.Certain constants are known to many decimal digits and recur throughout many diverse areas of mathematics, often in unexpected and surprising places (e.g., pi, e, and to some extent, the Euler-Mascheroni constant..

### Sexy primes

Sexy primes are pairs of primes of the form (, ), so-named since "sex" is the Latin word for "six.". The first few sexy prime pairs are (5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), ... (OEIS A023201 and A046117). As of November 2005, the largest known sexy prime pair starts with(1)where is a primorial. These primes have 10154 digits and were found by M. Fleuren, T. Alm, and J. K. Andersen (Andersen 2005).Sexy constellations also exist. The first few sexy triplets (i.e., numbers such that each of is prime but is not prime) are (7, 13, 19), (17, 23, 29), (31, 37, 43), (47, 53, 59), ... (OEIS A046118, A046119, and A046120). As of October 2005, the largest known sexy triplet starts with(2)These primes have 5132 digit digits and were found by Davis (2005).The first few sexy quadruplets are (11, 17, 23, 29), (41, 47, 53, 59), (61, 67, 73, 79), (251, 257, 263, 269),..

### Irregular prime

In a 1847 talk to the Académie des Sciences in Paris, Gabriel Lamé (1795-1870) claimed to have proven Fermat's last theorem. However, Joseph Liouville immediately pointed out an error in Lamé's result by pointing out that Lamé had incorrectly assumed unique factorization in the ring of -cyclotomic integers. Kummer had already studied the failure of unique factorization in cyclotomic fields and subsequently formulated a theory of ideals which was later further developed by Dedekind.Kummer was able to prove Fermat's last theorem for all prime exponents falling into a class he called "regular." "Irregular" primes are thus primes that are not a member of this class, and a prime is irregular iff divides the class number of the cyclotomic field generated by . Equivalently, but more conveniently, an odd prime is irregular iff divides the numerator of a Bernoulli number with .An infinite number..

### Consecutive number sequences

Consecutive number sequences are sequences constructed by concatenating numbers of a given type. Many of these sequences were considered by Smarandache and so are sometimes known as Smarandache sequences.The most obvious consecutive number sequence is the sequence of the first positive integers joined left-to-right, namely 1, 12, 123, 1234, ... (OEIS A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. 913). In this work, members of this sequence will be termed Smarandache numbers and the th such number written . No Smarandache primes exist for (Great Smarandache PRPrime search; Dec. 5, 2016).The th term of the "reverse integer sequence" consists of the concatenation of the first positive integers written right-to-left: 1, 21, 321, 4321, ... (OEIS A000422; Smarandache 1993, Dumitrescu and Seleacu 1994, Stephan 1998). The terms up to are given by (1)(2)The..

### Home prime

The prime reached starting from a number , concatenating its prime factors, and repeating until a prime is reached. For example, for ,so 311 is the home prime of 9. For , 3, ..., the first few are 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ... (OEIS A037274). Probabilistic arguments give exactly zero for the chance that the sequence of integers starting at a given number contains no prime, so a home prime should exist for every positive integer.Since prime numbers have trivial home primes (themselves), attention can be restricted to composite numbers. The numbers of steps to arrive at a home prime for composite numbers 4, 6, 8, 9, ... are 2, 1, 13, 2, 4, 1, 5, 4, 4, 1, 15, 1, ... (OEIS A037271), and the primes they reach are 211, 23, 3331113965338635107, 311, 773, 223, ... (OEIS A037272).The largest home prime for is , although its value is not known. The first few terms in the home prime sequence for 49 are 49, 77, 711, 3379, 31109, 132393, 344131, .....

### Repunit

A repunit is a number consisting of copies of the single digit 1. The term "repunit" was coined by Beiler (1966), who also gave the first tabulation of known factors.In base-10, repunits have the form(1)(2)Repunits therefore have exactly decimal digits. Amazingly, the squares of the repunits give the Demlo numbers, , , , ... (OEIS A002275 and A002477).The number of factors for the base-10 repunits for , 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS A046053). The base-10 repunit probable primes occur for , 19, 23, 317, and 1031, 49081, 86453, 109297, and 270343 (OEIS A004023; Madachy 1979, Williams and Dubner 1986, Ball and Coxeter 1987, Granlund, Dubner 1999, Baxter 2000), where is the largest proven prime (Williams and Dubner 1986). T. Granlund completed a search up to in 1998 using two months of CPU time on a parallel computer. The search was extended by Dubner (1999), culminating in the discovery of the probable prime ...

### Rectilinear crossing number

The rectilinear crossing number of a graph is the minimum number of crossings in a straight line drawing of in a plane. It is variously denoted , (Schaefer 2017), , or .It is sometimes claimed that the rectilinear crossing number is also known as the linear or geometric(al) crossing number, but evidence for that is slim (Schafer 2017).A disconnected graph has a rectilinear crossing number equal to the sums of the rectilinear crossing numbers of its connected components.When the (non-rectilinear) graph crossing number satisfies ,(1)(Bienstock and Dean 1993). While Bienstock and Dean don't actually prove equality for the case , they state it can be established analogously to . The result cannot be extended to , since there exist graphs with but for any (Bienstock and Dean 1993; Schaefer 2017, p. 54).G. Exoo (pers. comm., May 11-12, 2019) has written a program which can compute rectilinear crossing numbers for cubic graphs up to around..

### Heilbronn triangle problem

The Heilbronn triangle problem is to place points in a disk (square, equilateral triangle, etc.) of unit area so as to maximize the area of the smallest of the triangles determined by the points. For points, there is only a single triangle, so Heilbronn's problem degenerates into finding the largest triangle that can be constructed from points in a square. For , there are four possible triangles for each configuration, so the problem is to find the configuration of points for which the smallest of these four triangles is the maximum possible.For a unit square, the first few maxima of minimaltriangle areas are(1)(2)(3)(4)(5)(6)(7)(8)For larger values of , proofs of optimality are open, but the best known results are(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)with the configurations leading to maximum minimal triangles illustrated above (Friedman 2006; Comellas and Yebra 2002; D. Cantrell..

### Champernowne constant digits

The Champernowne constant has decimal expansion(OEIS A033307).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 1, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ... (OEIS A224896).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, ... (OEIS A229186).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 0, 00, 000, 0000, ..., which end at digits 11, 192, 2893, 38894, 488895, ... (OEIS A072290).The digit sequence 0123456789 first occurs at positions 11234567799, 22345677908, 33456779017, 44567790126, 55677901235, 66779012344, ... (OEIS A000000) and 9876543210 at positions 7777777779, 9876543212, 19987654323, 30998765434, 42099876545, 53209987656, 64320998767,..

### Heesch number

The Heesch number of a closed plane figure is the maximum number of times that figure can be completely surrounded by copies of itself. The determination of the maximum possible (finite) Heesch number is known as Heesch's problem. The Heesch number of a triangle, quadrilateral, regular hexagon, or any other shape that can tile or tessellate the plane, is infinity. Conversely, any shape with infinite Heesch number must tile the plane (Eppstein).A tile invented by R. Ammann has Heesch number three (Senechal 1995), and Mann has found an infinite family of tiles with Heesch number five (illustrated above), the largest (finite) number known.A database of Heesch tilings is maintained by Mann (2008).

### Pythagoras's constant digits

Pythagoras's constant has decimal expansion(OEIS A000129), It was computed to decimal digits by A. J. Yee on Feb. 9, 2012.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 114, 1481, 3308, 72459, 226697, 969836, 119555442, 2971094743, ... (OEIS A224871).-constant primes occur at 55, 97, 225, 11260, 11540, ... (OEIS A115377) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 14, 1, 5, 7, 2, 8, 9, 12, 19, ... (OEIS A229199).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 81, 748, 8505, 30103, 489568, ... (OEIS A000000), which end at digits 19, 420, 8326, 94388, 1256460, 13043524, ... (OEIS A000000).The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does, starting..

### Catalan's constant digits

Based on methods developer in collaboration with M. Leclert, Catalan (1865) computed the constant(OEIS A006752) now known as Catalans' constant to 9 decimals. In 1867, M. Bresse subsequently computed to 24 decimal places using a technique from Kummer. Glaisher evaluated to 20 (Glaisher 1877) and subsequently 32 decimal digits (Glaisher 1913). Catalan's constant was computed to decimal digits by A. Roberts on Dec. 13, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for Catalan's constant is given for , 2, ... by 2, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ... (OEIS A224819).-constant primes occur for 52, 276, 25477, ... (OEIS A118328) digits.It is not known if is normal, but the following table giving the counts of digits in the first terms shows that the decimal digits are very uniformly distributed up to at least .OEIS101000A22461506989769828996209997849998686999960671A2246162189410399832996971000293100038131000063052A22469601093980100781001681001789100051221000008063A22470607104101498599958099967299956761000014834A22471711110796110051100074100016599953771000018715A2247743108910031006210005399996599993091000007776A22477511278985998610020199871210000674999988167A2248160111241032100281000831000510100038631000005768A2248170310210581019210035299929899974371000008639A224818312111952100841001729998121000004399992436The..

### Probable prime

A number satisfying Fermat's little theorem (or some other primality test) for some nontrivial base. A probable prime which is shown to be composite is called a pseudoprime (otherwise, of course, it is a prime).As of Sep. 2013, the largest known probable primes are the Wagstaff primes and , both found by R. Propper in Sep. 2013 and which have and decimal digits, respectively. Other large known probable primes are the "dual Sierpinski numbers" (Moore 2009) given by and , which have and decimal digits, respectively (Lifchitz and Lifchitz).

### Prime triplet

A prime triplet is a prime constellation of the form (, , ), (, , ), etc. Hardy and Wright (1979, p. 5) conjecture, and it seems almost certain to be true, that there are infinitely many prime triplets of the form (, , ) and (, , ).tripletSloanefirst member(, , )A0220045, 11, 17, 41, 101, 107, ...(, , )A0461343, 5, 11, 29, 59, 71, 101, ...(, , )A0461355, 11, 17, 29, 41, 59, 71, ...(, , )A0220057, 13, 37, 67, 97, 103, ...(, , )A0461363, 7, 13, 19, 37, 43, 79, ...(, , )A0461377, 19, 67, 97, 127, 229, ...(, , )A0461385, 11, 23, 53, 101, 131, ...(, , )A0461397, 13, 31, 37, 61, 73, 97, ...(, , )A0232415, 7, 11, 17, 31, 41, 47, ...(, , )A0461415, 11, 29, 59, 71, 89, 101, ...As of Apr. 2019, the largest known prime triplet of the form has smallest memberand each of its three members has decimal digits...

### Carmichael number

A Carmichael number is an odd composite number which satisfies Fermat's little theorem(1)for every choice of satisfying (i.e., and are relatively prime) with . A Carmichael number is therefore a pseudoprime to any base. Carmichael numbers therefore cannot be found to be composite using Fermat's little theorem. However, if , the congruence of Fermat's little theorem is nonzero, thus identifying a Carmichael number as composite.Carmichael numbers are sometimes called "absolute pseudoprimes" and also satisfy Korselt's criterion. R. D. Carmichael first noted the existence of such numbers in 1910, computed 15 examples, and conjectured that there were infinitely many. In 1956, Erdős sketched a technique for constructing large Carmichael numbers (Hoffman 1998, p. 183), and a proof was given by Alford et al. (1994).Any solution to Lehmer's totient problemmust be a Carmichael number.The first few Carmichael..

### Golomb ruler

An -mark Golomb ruler is a set of distinct nonnegative integers , called "marks," such that the positive differences , computed over all possible pairs of different integers , ..., with are distinct.Let be the largest integer in an -mark Golomb ruler. Then an -mark Golomb ruler is optimal if 1. There exists no other -mark Golomb rulers having smaller largest mark , and 2. The ruler is written in canonical form as the "smaller" of the equivalent rulers and , where "smaller" means the first differing entry is less than the corresponding entry in the other ruler. In such a case, is the called the "length" of the optimal -mark ruler.Thus, (0, 1, 3) is the unique optimal 3-mark Golomb ruler modulo reversal (i.e., (0, 2, 3) is considered the same ruler).For example, the set (0, 1, 3, 7) is 4-mark Golomb ruler since its differences are (, , , , , ), all of which are distinct. However, the unique optimal Golomb 4-mark ruler..

### Brun's constant

The number obtained by adding the reciprocals of the odd twinprimes,(1)By Brun's theorem, the series converges to a definite number, which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1989, p. 201). By contrast, the series of all prime reciprocals diverges to infinity, as follows from the Mertens second theorem by letting (which provides a stronger characterization of the divergence than Euler's proof that , obtained more than a century before Mertens' proof).Shanks and Wrench (1974) used all the twin primes among the first 2 million numbers. Brent (1976) calculated all twin primes up to 100 billion and obtained (Ribenboim 1989, p. 146)(2)assuming the truth of the first Hardy-Littlewood conjecture. Using twin primes up to , Nicely (1996) obtained(3)(Cipra 1995, 1996), in the process discovering a bug in Intel's® PentiumTM microprocessor. Using twin primes up to , Nicely..

### Prime gaps

A prime gap of length is a run of consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes and bounding a prime gap of length is , where is the th prime number. Since the prime difference function(1)is always even (except for ), all primes gaps are also even. The notation is commonly used to denote the smallest prime corresponding to the start of a prime gap of length , i.e., such that is prime, , , ..., are all composite, and is prime (with the additional constraint that no smaller number satisfying these properties exists).The maximal prime gap is the length of the largest prime gap that begins with a prime less than some maximum value . For , 2, ..., is given by 4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ... (OEIS A053303).Arbitrarily large prime gaps exist. For example, for any , the numbers , , ..., are all composite (Havil 2003, p. 170). However, no general method..

### Brocard's problem

Brocard's problem asks to find the values of for which is a square number , where is the factorial (Brocard 1876, 1885). The only known solutions are , 5, and 7. Pairs of numbers are called Brown numbers. In 1906, Gérardin claimed that, if , then must have at least 20 digits. Unaware of Brocard's query, Ramanujan considered the same problem in 1913. Gupta (1935) stated that calculations of up to gave no further solutions.It is virtually certain that there are no more solutions (Guy 1994). In fact, Dabrowski (1996) has shown that has only finitely many solutions for general , although this result requires assumption of a weak form of the abc conjecture if is square).There are no other solutions with (Wells 1986, p. 70), and Berndt and Galway have further searched up to without finding any further solutions.Wilson has also computed the least such that is square starting at , giving 1, 1, 3, 1, 9, 27, 15, 18, 288, 288, 420, 464, 1856, ... (OEIS..

### Prime factorization algorithms

Many algorithms have been devised for determining the prime factors of a given number (a process called prime factorization). They vary quite a bit in sophistication and complexity. It is very difficult to build a general-purpose algorithm for this computationally "hard" problem, so any additional information that is known about the number in question or its factors can often be used to save a large amount of time.The simplest method of finding factors is so-called "direct search factorization" (a.k.a. trial division). In this method, all possible factors are systematically tested using trial division to see if they actually divide the given number. It is practical only for very small numbers.The fastest-known fully proven deterministic algorithm is the Pollard-Strassen method(Pomerance 1982; Hardy et al. 1990)...

### Golden ratio digits

The golden ratio has decimal expansion(OEIS A001622). It can be computed to digits of precision in 24 CPU-minutes on modern hardware and was computed to decimal digits by A. J. Yee on Jul. 8, 2010.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 62, 158, 1216, 72618, 2905357, 7446157, 41398949, 1574998166, ... (OEIS A224844).The digit sequence 0123456789 does not occur in the first digits of , but 9876543210 does, starting at position (E. Weisstein, Jul. 22, 2013).Phi-primes, i.e., -constant primes occur for 7, 13, 255, 280, 97241, ... (OEIS A064119) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 5, 1, 20, 6, 12, 23, 2, 11, 4, 8, 232, ... (OEIS A088577).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit..

### Bitwin chain

A bitwin chain of length one consists of two pairs of twinprimes with the property that they are related by being of the form:(1)The first few values of generating bitwin chains are 6, 30, 660, 810, 2130, 2550, 3330, ... (OEIS A066388).In general a chain of length consists of pairs of twin primes,(2)Bitwin chains can also be viewed as consisting of two related Cunninghamchains of the first and second kinds,(3)P. Jobling (1999) found the largest known chain of length six,(4)where to 6.

### Prime counting function

The prime counting function is the function giving the number of primes less than or equal to a given number (Shanks 1993, p. 15). For example, there are no primes , so . There is a single prime (2) , so . There are two primes (2 and 3) , so . And so on.The notation for the prime counting function is slightly unfortunate because it has nothing whatsoever to do with the constant . This notation was introduced by number theorist Edmund Landau in 1909 and has now become standard. In the words of Derbyshire (2004, p. 38), "I am sorry about this; it is not my fault. You'll just have to put up with it."Letting denote the th prime, is a right inverse of since(1)for all positive integers. Also,(2)iff is a prime number.The first few values of for , 2, ... are 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, ... (OEIS A000720). The Wolfram Language command giving the prime counting function for a number is PrimePi[x], which works up to a maximum value of .The notation..

### Goldbach conjecture

Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes" (Goldbach 1742; Dickson 2005, p. 421). Note that Goldbach considered the number 1 to be a prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes. Two primes such that for a positive integer are sometimes called a Goldbach partition (Oliveira e Silva).According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and..

### Bernoulli number

The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function(1)These numbers arise in the series expansions of trigonometric functions, and areextremely important in number theory and analysis.There are actually two definitions for the Bernoulli numbers. To distinguish them, the Bernoulli numbers as defined in modern usage (National Institute of Standards and Technology convention) are written , while the Bernoulli numbers encountered in older literature are written (Gradshteyn and Ryzhik 2000). In each case, the Bernoulli numbers are a special case of the Bernoulli polynomials or with and .The Bernoulli number and polynomial should not be confused with the Bell numbers and Bell polynomial, which are also commonly denoted and , respectively.Bernoulli numbers defined by the modern definition are denoted and sometimes called "even-index" Bernoulli numbers...

### Belphegor prime

A Belphegor prime (also known as a Beelphegor prime) is a prime Belphegor number, i.e., a palindromic prime of the form . The first few Belphegor primes are the Belphegor numberswith , 13, 42, 506, 608, 2472, 2623, 28291, 181298, ... (OEIS A232448). Shifting this sequence by one gives 1, 14, 43, 507, 609, ... (OEIS A156166).The above definition generalizes the original definition due to Pickover, who named"the" Belphegor prime after one of the Seven Princes of Hell who is the demon of inventiveness. This prime has a number of numerological properties, including a central beast number 666 which is surrounded on each side by 13 (a number traditionally associated with bad luck) zeros and an overall decimal number length of 31-which is 13 backwards.

### Prime arithmetic progression

An arithmetic progression of primes is a set of primes of the form for fixed and and consecutive , i.e., . For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.It had long been conjectured that there exist arbitrarily long sequences of primes in arithmetic progression (Guy 1994). As early as 1770, Lagrange and Waring investigated how large the common difference of an arithmetic progression of primes must be. In 1923, Hardy and Littlewood (1923) made a very general conjecture known as the k-tuple conjecture about the distribution of prime constellations, which includes the hypothesis that there exist infinitely long prime arithmetic progressions as a special case. Important additional theoretical progress was subsequently made by van der Corput (1939), who proved than there are infinitely many triples of primes in arithmetic progression, and Heath-Brown (1981),..

### Gilbreath's conjecture

Let the difference of successive primes be defined by , and by(1)N. L. Gilbreath claimed that for all (Guy 1994). In 1959, the claim was verified for . In 1993, Odlyzko extended the claim to all primes up to .Gilbreath's conjecture is equivalent to the statement that, in the triangular array of the primes, iteratively taking the absolute difference of each pair of terms(2)(OEIS A036262), always gives leading term 1(after the first row).The number of terms before reaching the first greater than two in the second, third,etc., rows are given by 3, 8, 14, 14, 25, 23, 22, 25, ... (OEIS A000232).

### Ap&eacute;ry's constant digits

Apéry's constant is defined by(OEIS A002117) where is the Riemann zeta function. was computed to decimal digits by E. Weisstein on Sep. 16, 2013.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 10, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ... (OEIS A229074).-constant prime occur for , 55, 109, 141, ... (OEIS A119334), corresponding to the primes 1202056903, 1202056903159594285399738161511449990764986292340498881, ... (OEIS A119333).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 3, 1, 2, 10, 16, 6, 7, 23, 18, 8, ... (OEIS A229187).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 89, 211, 2861, 43983, 292702, 8261623, ... (OEIS A036902), which end at digits 23, 457, 7839, 83054, 1256587,..

### Generalized fermat number

There are two different definitions of generalized Fermat numbers, one of which is more general than the other. Ribenboim (1996, pp. 89 and 359-360) defines a generalized Fermat number as a number of the form with , while Riesel (1994) further generalizes, defining it to be a number of the form . Both definitions generalize the usual Fermat numbers . The following table gives the first few generalized Fermat numbers for various bases .OEISgeneralized Fermat numbers in base 2A0002153, 5, 17, 257, 65537, 4294967297, ...3A0599194, 10, 82, 6562, 43046722, ...4A0002155, 17, 257, 65537, 4294967297, 18446744073709551617, ...5A0783036, 26, 626, 390626, 152587890626, ...6A0783047, 37, 1297, 1679617, 2821109907457, ...Generalized Fermat numbers can be prime only for even . More specifically, an odd prime is a generalized Fermat prime iff there exists an integer with and (Broadhurst 2006).Many of the largest known prime numbers are generalized..

### Ap&eacute;ry's constant

Apéry's constant is defined by(1)(OEIS A002117) where is the Riemann zeta function. Apéry (1979) proved that is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of (Hata 2000). arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.The following table summarizes progress in computing upper bounds on the irrationality measure for . Here, the exact values for is given by(2)(3)(Hata 2000).upper boundreference15.513891Rhin and Viola (2001)28.830284Hata (1990)312.74359Dvornicich and Viola (1987)413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)Beukers (1979) reproduced Apéry's rational approximation to using the triple..

### Amicable pair

An amicable pair consists of two integers for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other. Amicable pairs are occasionally called friendly pairs (Hoffman 1998, p. 45), although this nomenclature is to be discouraged since the numbers more commonly known as friendly pairs are defined by a different, albeit related, criterion. Symbolically, amicable pairs satisfy(1)(2)where(3)is the restricted divisor function. Equivalently, an amicable pair satisfies(4)where is the divisor function. The smallest amicable pair is (220, 284) which has factorizations(5)(6)giving restricted divisor functions(7)(8)(9)(10)The quantity(11)in this case, , is called the pair sum. The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), ... (OEIS A002025 and A002046). An exhaustive..

### Aliquot sequence

Letwhere is the divisor function and is the restricted divisor function. Then the sequence of numbersis called an aliquot sequence. If the sequence for a given is bounded, it either ends at or becomes periodic. 1. If the sequence reaches a constant, the constant is known as a perfect number. A number that is not perfect, but for which the sequence becomes constant, is known as an aspiring number. 2. If the sequence reaches an alternating pair, it iscalled an amicable pair. 3. If, after iterations, the sequence yields a cycle of minimum length of the form , , ..., , then these numbers form a group of sociable numbers of order . The lengths of the aliquot sequences for , 2, ... are 1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, ... (OEIS A044050).It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. Guy (1994) cites the largest computed value as , though this has since been extended..

### Fibonacci prime

A Fibonacci prime is a Fibonacci number that is also a prime number. Every that is prime must have a prime index , with the exception of . However, the converse is not true (i.e., not every prime index gives a prime ).The first few (possibly probable) prime Fibonacci numbers are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (OEIS A005478), corresponding to indices , 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, ... (OEIS A001605). (Note that Gardner's statement that is prime (Gardner 1979, p. 161) is incorrect, especially since 531 is not even prime, which it must be for to be prime.) The following table summarizes Fibonacci (possibly probable) primes with index .termindexdigitsdiscovererstatus2453871126proven prime; https://primes.utm.edu/primes/page.php?id=511292593111946proven prime; https://primes.utm.edu/primes/page.php?id=374702696772023proven prime; https://primes.utm.edu/primes/page.php?id=3553727144313016proven..

### Fermat prime

A Fermat prime is a Fermat number that is prime. Fermat primes are therefore near-square primes.Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88). At present, however, the only Fermat numbers for for which primality or compositeness has been established are all composite.The only known Fermat primes are(1)(2)(3)(4)(5)(OEIS A019434), and it seems unlikely that any more will be found using current computational methods and hardware. It follows that is prime for the special case together with the Fermat prime indices, giving the sequence 2, 3, 5, 17, 257, and 65537 (OEIS A092506). is a Fermat prime if and only if the period length of is equal to . In other words, Fermat primes are full reptend primes...

### 10

The number 10 (ten) is the basis for the decimal system of notation. In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the s place. For example, the number 1234.56 specifies(1)The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, ... (OEIS A011557), called one, ten, hundred, thousand, ten thousand, hundred thousand, million, 10 million, 100 million, and so on. The names of subsequent decimal places for large numbers differ depending on country. Any power of 10 which can be written as the product of two numbers not containing 0s must be of the form for an integer such that neither nor contains any zeros. The largest known such number is(2)A complete list of such known numbers is(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(Madachy 1979). Since..

### Soldner's constant continued fraction

The continued fraction for is given by [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...] (OEIS A099803).The positions at which the numbers 1, 2, ... occur in the continued fraction are0, 1, 6, 2, 47, 28, 21, 107, 114, ... (OEIS A000000).The high-water marks are 1, 2, 4, 47, 99, 294, 527, 616, 1152, ... (OEIS A099804), which occur at positions 0, 1, 2, 11, 69, 125, 201, 584, 1591, 2435, ... (OEIS A229230).Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

### E continued fraction

The simple continued fraction representations of given by [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] (OEIS A003417). This continued fraction is sometimes known as Euler's continued fraction. A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The convergents can be given in closed form as ratios of confluent hypergeometric functions of the first kind (Komatsu 2007ab), with the first few being 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, ... (OEIS A007676 and A007677). These are good to 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, ... (OEIS A114539) decimal digits, respectively.Other continued fraction representations are(1)(2)(3)(Olds 1963, pp. 135-136). Amazingly, not only the continued fractions , but those of rational powers of show regularity, for example(4)(5)(6)(7)A beautiful non-simple continued fraction for is given by(8)(Wall 1948, p. 348).Let the continued fraction of be denoted..

### Pi continued fraction

The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The first few convergents are 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.The very large term 292 means that the convergent(1)is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). A nice expression for the third convergent of is given by(2)(Stoschek).The Engel expansion of is 1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ... (OEIS A006784).The following table summarizes some record computations of the continued fraction of pi.termsdatereference1977W. Gosper..

### Khinchin's constant continued fraction

The continued fraction for is [2; 1, 2, 5, 1, 1, 2, 1, 1, ...] (OEIS A002211). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The convergents are 2, 3, 8/3, 43/16, 51/19, ... (OEIS A127005and A127006).The incrementally largest terms are 2, 5, 10, 24, 90, 770, ... (OEIS A054866), which occur at positions 0, 3, 10, 15, 23, 104, 1701, ... (OEIS A224852; illustrated above).The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 0, 9, 46, 3, 33, 75, 64, 118, 10, 103, 26, 102, 109, ... (OEIS A224851). The smallest number not occurring in the first terms of the continued fraction are 236, 260, 265, 279, 282, ... (E. Weisstein, Jul. 22, 2013).

### Khinchin's constant

Let(1)be the simple continued fraction of a "generic" real number , where the numbers are the partial quotients. Khinchin (1934) considered the limit of the geometric mean(2)as . Amazingly, except for a set of measure 0, this limit is a constant independent of given by(3)(OEIS A002210), as proved in Kac (1959).The constant is known as Khinchin's constant, and is commonly also spelled "Khintchine'sconstant" (Shanks and Wrench 1959, Bailey et al. 1997).It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.It is not known if is irrational, let alone transcendental.While it is known that almost all numbers have limits that approach , this fact has not been proven for any explicit real number , e.g., a real number cast in terms of fundamental constants..

### Champernowne constant continued fraction

The first few terms in the continued fraction of the Champernowne constant are [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 45754...10987, 6, 1, 1, 21, ...] (OEIS A030167), and the number of decimal digits in these terms are 0, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 166, 1, ... (OEIS A143532). E. W. Weisstein computed terms of the continued fraction on Jun. 30, 2013 using the Wolfram Language.First occurrences of the terms 1, 2, 3, ... in the continued fraction occur at , 28, 13, 9, 93, 20, 31, 2, 3, 339, 71, 126, 107, ... (OEIS A038706). The smallest unknown value is 188, which has .The continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the size of the continued fraction high-water marks display apparent patterns (Sikora 2012). Large terms greater than occur at positions 5, 19, 41, 102, 163, 247, 358, 460, ... and have 6, 166, 2504, 140, 33102, 109, 2468,..

### Catalan's constant continued fraction

The simple continued fraction representations for Catalan's constant is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (OEIS A014538). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.Record computations are summarized below.termsdatebyJul. 20, 2013E. WeissteinAug. 7, 2013E. WeissteinThe plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 13, 14, 7, 45, 36, 10, 4, 21, 2, ... (OEIS A196461; illustrated above). The smallest number not occurring in the first terms of the continued fraction are 31516, 31591, 32600, 32806, 33410, ... (E. Weisstein, Aug. 8, 2013).The cumulative largest terms in the continued fraction are 0, 1, 10, 88, 322, 330, 1102, 6328, ... (OEIS A099789), which occur at positions 0, 1, 2, 6, 105, 284, 747, 984, 2230, 5377, ... (OEIS A099790).Let the continued fraction..

### Ap&eacute;ry's constant continued fraction

The continued fraction for Apéry's constant is [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] (OEIS A013631).The positions at which the numbers 1, 2, ... occur in the continued fraction are 0, 11, 24, 1, 63, 26, 16, 139, 9, 118, 20, ... (OEIS A229057). The incrementally maximal terms are 1, 4, 18, 30, 428, 458, 527, ... (OEIS A033166), which occur at positions 0, 1, 3, 28, 62, 571, 1555, 2012, 2529, ... (OEIS A229055).Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

### Natural logarithm of 10 continued fraction

The continued fraction for is [0; 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, ...] (OEIS A016730).The Engel expansion is 2, 3, 7, 9, 104, 510, 1413,... (OEIS A059180).The incrementally largest terms in the continued fraction of are 2, 3, 6, 26, 716, 774, 982, 1324, 4093, 10322, ... (OEIS A228346), which occur at positions 0, 1, 7, 17, 30, 136, 962, 1163, 1261, 1293, ... (OEIS A228345).The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 4, 0, 1, 11, 18, 7, 44, 159, 74, 212, 260, 182, 43, 152, 59, 84, 40, 86, 27, 89, ... (OEIS A228270). The smallest number not occurring in the first terms of the continued fraction are 40230, 45952, 46178, 46530, ... (E. Weisstein, Aug. 18, 2013).Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left..

### Natural logarithm of 2 continued fraction

The continued fraction for is [0; 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, ...] (OEIS A016730). It has been computed to terms by E. Weisstein (Aug. 21, 2013).The Engel expansion is 2, 3, 7, 9, 104, 510, 1413,... (OEIS A059180).The incrementally largest terms in the continued fraction are 0, 1, 2, 3, 6, 10, 13, 14, ... (OEIS A120754), which occur at positions 0, 1, 2, 3, 5, 15, 28, ... (OEIS A120755).The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 2, 3, 30, 40, 5, 29, 89, 88, 15, ... (OEIS A228269). The smallest number not occurring in the first terms of the continued fraction are 42112, 42387, 43072, 45089, ... (E. Weisstein, Aug. 21, 2013).Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure)..

### Computational number theory

Computational number theory is the branch of number theory concerned with finding and implementing efficient computer algorithms for solving various problems in number theory. Much progress has been made in this field in recent years, both in terms of improved computer speed and in terms of finding more efficient algorithms. Two important applications of computational number theory are primality testing and prime factorization of large integers.Primality testing is considered easy in the sense that very large general numbers (currently up to 4000 digits or so) can be tested reliably for primality. In fact, on August 6, 2002, Agrawal, Saxena, and Kayal found a polynomial time algorithm for testing and proving the primality of general numbers. Although this algorithm is still impractical, it was a landmark discovery, since polynomial time algorithms are considered easy. On the other hand, factoring is considered hard in the sense that..

### Integer sequence primes

Just as many interesting integer sequences can be defined and their properties studied, it is often of interest to additionally determine which of their elements are prime. The following table summarizes the indices of the largest known prime (or probable prime) members of a number of named sequences.sequenceOEISdigitsdiscoverersearch limitcommentsalternating factorialA00127259961260448M. Rodenkirch (Sep. 18, 2017)100000 (M. Rodenkirch, Dec. 15, 2017)finite sequence; largest certified prime has index 661; the rest are probable primesApéry-constant primeA119334141141E. W. Weisstein (May 14, 2006)9089 (E. W. Weisstein, Mar. 22, 2008)status unknownApéry number A092825662410136E. W. Weisstein (Mar. 2004) (E. W. Weisstein, Mar. 2004)probable primeApéry number 87E. W. Weisstein..

### Square packing

Find the minimum size square capable of bounding equal squares arranged in any configuration. The first few cases are illustrated above (Friedman). The only packings which have been proven optimal are 2, 3, 5, 6, 7, 8, 14, 15, 24, and 35, in addition to the trivial cases of the square numbers (Friedman).If for some , it is conjectured that the size of the minimum bounding square is for small . The smallest for which the conjecture is known to be violated is (with ).The following table gives the smallest known side lengths for a square into which unit squares can be packed (Friedman 2005). An asterisk (*)indicates that a packing has been proven to be optimal.exactapprox.exactapprox.1*1116*442*22174.6755...3*22184.822...4*22194.885...5*2.707...20556*3321557*3322558*3323*559*3324*5510*3.707...25*55113.877...265.6214...1244275.7072...1344285.8285...14*44295.9465...15*44The best known packings of squares into a circle..

### Circle packing

A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres is called a sphere packing. Tessellations of regular polygons correspond to particular circle packings (Williams 1979, pp. 35-41). There is a well-developed theory of circle packing in the context of discrete conformal mapping (Stephenson).The densest packing of circles in the plane is the hexagonal lattice of the bee's honeycomb (right figure; Steinhaus 1999, p. 202), which has a packing density of(1)(OEIS A093766; Wells 1986, p. 30). Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fejes Tóth proved that the hexagonal lattice is indeed the densest of all possible plane packings.Surprisingly, the circular disk is not the least economical region for packing the plane. The "worst"..

### Kissing number

The number of equivalent hyperspheres in dimensions which can touch an equivalent hypersphere without any intersections, also sometimes called the Newton number, contact number, coordination number, or ligancy. Newton correctly believed that the kissing number in three dimensions was 12, but the first proofs were not produced until the 19th century (Conway and Sloane 1993, p. 21) by Bender (1874), Hoppe (1874), and Günther (1875). More concise proofs were published by Schütte and van der Waerden (1953) and Leech (1956). After packing 12 spheres around the central one (which can be done, for example, by arranging the spheres so that their points of tangency with the central sphere correspond to the vertices of an icosahedron), there is a significant amount of free space left (above figure), although not enough to fit a 13th sphere.Exact values for lattice packings are known for to 9 and (Conway and Sloane 1993, Sloane and..

### Busy beaver

A busy beaver is an -state, 2-color Turing machine which writes a maximum number of 1s before halting (Rado 1962; Lin and Rado 1965; Shallit 1998). Alternatively, some authors define a busy beaver as a Turing machine that performs a maximum number of steps when started on an initially blank tape before halting (Wolfram 2002, p. 889). The process leading to the solution of the three-state machine is discussed by Lin and Rado (1965) and the process leading to the solution of the four-state machine is discussed by Brady (1983) and Machlin and Stout (1990).For , 2, ..., (also known as Rado's sigma function) is given by 1, 4, 6, 13, ... (OEIS A028444; Rado 1962; Wolfram 2002, p. 889). The next few terms are not known, but examples have been constructed giving lower limits of and (Marxen). The illustration above shows a 3-state Turing machine with maximal (Lin and Rado 1965, Shallit 1998).The maximum number of steps which an -state 2-color Turing..

### Factorial prime

A factorial prime is a prime number of the form , where is a factorial. is prime for , 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 26951, 34790, 94550, 103040, 147855, 208003, ... (OEIS A002982), the largest of which are summarized in the following table.digitsdiscoverer107,707Marchal, Carmody, and Kuosa (Caldwell; May 2002)142,891Marchal, Carmody, and Kuosa (Caldwell; May 2002)429,390D. Domanov/PrimeGrid (Oct. 4, 2010)471,794J. Winskill/PrimeGrid (Dec. 14, 2010)700,177PrimeGrid (Aug. 30, 2013)1,015,843S. Fukui (Jul. 25, 2016; https://primes.utm.edu/primes/page.php?id=121944) is prime for , 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ... (OEIS A002981; Wells 1986, p. 70), the largest of which are summarized in the following table.digitsdiscoverer107,707K. Davis..

### Double factorial prime

A double factorial prime is a prime number of the form , where is a double factorial. is prime for , 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, ... (OEIS A007749), the largest of which are summarized in the following table.digitsdiscoverer169,435S. Fukai (Jun. 5, 2015)229,924S. Fukai (Jun. 5, 2015)344,538S. Fukai (Apr. 21, 2016) is prime for , 1, 2, 518, 33416, 37310, 52608, 123998, ... (OEIS A080778), the largest of which are summarized in the following table.digitsdiscoverer112,762H. Jamke (Jan. 3, 2008)288,864S. Fukai (Jun. 5, 2015)

### Gaussian prime

Gaussian primes are Gaussian integers satisfying one of the following properties. 1. If both and are nonzero then, is a Gaussian prime iff is an ordinary prime. 2. If , then is a Gaussian prime iff is an ordinary prime and . 3. If , then is a Gaussian prime iff is an ordinary prime and . The above plot of the complex plane shows the Gaussianprimes as filled squares.The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ... (OEIS A002145). The Gaussian primes with are given by , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3, , , , , , , , , .The numbers of Gaussian primes with complex modulus (where the definition has been used) for , 1, ... are 0, 100, 4928, 313752, ... (OEIS A091134).The cover of Bressoud and Wagon (2000) shows an illustration of the distributionof Gaussian primes in the complex plane.As of 2009, the largest known Gaussian prime, found in Sep. 2006, is , whose real and imaginary parts both have decimal digits and whose squared..

### Folding

There are many mathematical and recreational problems related to folding. Origami,the Japanese art of paper folding, is one well-known example.It is possible to make a surprising variety of shapes by folding a piece of paper multiple times, making one complete straight cut, then unfolding. For example, a five-pointed star can be produced after four folds (Demaine and Demaine 2004, p. 23), as can a polygonal swan, butterfly, and angelfish (Demaine and Demaine 2004, p. 29). Amazingly, every polygonal shape can be produced this way, as can any disconnected combination of polygonal shapes (Demaine and Demaine 2004, p. 25). Furthermore, algorithms for determining the patterns of folds for a given shape have been devised by Bern et al. (2001) and Demaine et al. (1998, 1999).Wells (1986, p. 37; Wells 1991) and Gurkewitz and Arnstein (2003, pp. 49-59) illustrate the construction of the equilateral triangle, regular..

### Prime number

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer that has no positive integer divisors other than 1 and itself. More concisely, a prime number is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization ), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as the Gaussian primes.The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright..

### Ferrier's prime

According to Hardy and Wright (1979), the 44-digit Ferrier's primedetermined to be prime using only a mechanical calculator, is the largest prime found before the days of electronic computers. The Wolfram Language can verify primality of this number in a (small) fraction of a second, showing how far the art of numerical computation has advanced in the intervening years. It can be shown to be a probable prime almost instantaneously In:= FerrierPrime = (2^148 + 1)/17; In:= PrimeQ[FerrierPrime] // Timing Out= {0.01 Second, True}and verified to be an actual prime complete with primalitycertificate almost as quickly In:= <<PrimalityProving` In:= ProvablePrimeQ[FerrierPrime, "Certificate" -> True] // Timing Out= {0.04 Second,{True, {20988936657440586486151264256610222593863921,17, {2,{3,2,{2}},{5,2,{2}},{7,3,{2,{3,2,{2}}}}, {13,2,{2,{3,2,{2}}}},{19, 2,{2,{3,2,{2}}}},{37,2,{2,{3,2,{2}}}},{73,5,{..

### Prime array

Find the array of single digits which contains the maximum possible number of primes, where allowable primes may lie along any horizontal, vertical, or diagonal line.For the array, 11 primes are maximal and are contained in the two distinct arrays(1)giving the primes (3, 7, 13, 17, 31, 37, 41, 43,47, 71, 73) and (3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97), respectively.The best array is(2)which contains 30 primes: 3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 97, 113, 157, 179, ... (OEIS A032529). This array was found by Rivera and Ayala and shown by Weisstein in May 1999 to be maximal and unique (modulo reflection and rotation).The best arrays known are(3)all of which contain 63 primes. The first was found by C. Rivera and J. Ayala in 1998, and the other three by James Bonfield on April 13, 1999. Mike Oakes proved by computation that the 63 primes is optimal for the array.The best prime arrays known are(4)each of which contains 116 primes...

### Pi digits

has decimal expansion given by(1)(OEIS A000796). The following table summarizes some record computations of the digits of .1999Kanada, Ushio and KurodaDec. 2002Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003)Aug. 2012A. J. Yee (Yee)Aug. 2012S. Kondo and A. J. Yee (Yee)Dec. 2013A. J. Yee and S. Kondo (Yee)The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 places. Although he did not live to publish his result, it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of other calculations. The calculation of also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship..

### 13th root

Calculating the 13th root of a large number (that is a perfect 13th power) is a famous mental calculation challenge. However, because of difficulties in standardizing the time taken to find the root, the Guinness Book of World Records no longer maintains an entry for the 13th root.The official record (as of August 2005) is 13.55 seconds for a 100-digit perfect 13th power, as calculated by Alexis Lemaire on May 10, 2002. (By comparison, the Wolfram Language computes such roots in about 51 microseconds.) The record for a 200-digit number was also set by Lemaire, at 267.77 seconds (whereas the Wolfram Language takes roughly 82 microseconds).

### Collatz problem

A problem posed by L. Collatz in 1937, also called the mapping, problem, Hasse's algorithm, Kakutani's problem, Syracuse algorithm, Syracuse problem, Thwaites conjecture, and Ulam's problem (Lagarias 1985). Thwaites (1996) has offered a £1000 reward for resolving the conjecture. Let be an integer. Then one form of Collatz problem asks if iterating(1)always returns to 1 for positive . (If negative numbers are included, there are four known cycles (excluding the trivial 0 cycle): (4, 2, 1), (, ), (, , , , ), and (, , , , , , , , , , , , , , , , , ).)The members of the sequence produced by the Collatz are sometimes known as hailstone numbers. Conway proved that the original Collatz problem has no nontrivial cycles of length . Lagarias (1985) showed that there are no nontrivial cycles with length . Conway (1972) also proved that Collatz-type problems can be formally undecidable. Kurtz and Simon (2007) proved that a natural generalization of the..

### Smarandache sequences

Smarandache sequences are any of a number of simply generated integer sequences resembling those considered in published works by Smarandache such as the consecutive number sequences and Euclid numbers (Iacobescu 1997). Some other "Smarandache" sequences are given below.1. The concatenation of copies of the integer : 1, 22, 333, 4444, 55555, ... (OEIS A000461; Marimutha 1997). For , they have the simple formula(1)where is a repunit. In general,(2)where is the number of digits in . Since the th term is always divisible by , numbers in this sequences can never be prime. 2. The concatenation of the first Fibonacci numbers: 1, 11, 112, 1123, 11235, ... (OEIS A019523; Marimutha 1997). 3. The smallest number that is the sum of squares of two distinct earlier terms: 1, 2, 5, 26, 29, 677, ... (OEIS A008318; Bencze 1997). 4. The smallest number that is the sum of squares of any number of distinct earlier terms: 1, 1, 2, 4, 5, 6, 16, 17, ... (OEIS A008319;..

### Harmonic number

A harmonic number is a number of the form(1)arising from truncation of the harmonic series.A harmonic number can be expressed analytically as(2)where is the Euler-Mascheroni constant and is the digamma function.The first few harmonic numbers are 1, , , , , ... (OEIS A001008 and A002805). The numbers of digits in the numerator of for , 1, ... are 1, 4, 41, 434, 4346, 43451, 434111, 4342303, 43428680, ... (OEIS A114467), with the corresponding number of digits in the denominator given by 1, 4, 40, 433, 4345, 43450, 434110, 4342302, 43428678, ... (OEIS A114468). These digits converge to what appears to be the decimal digits of (OEIS A002285).The first few indices such that the numerator of is prime are given by 2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, ... (OEIS A056903). The search for prime numerators has been completed up to by E. W. Weisstein (May 13, 2009), and the following table summarizes the largest known values.decimal digitsdiscoverer6394227795E. W. Weisstein..

### Perfect number

Perfect numbers are positive integers such that(1)where is the restricted divisor function (i.e., the sum of proper divisors of ), or equivalently(2)where is the divisor function (i.e., the sum of divisors of including itself). For example, the first few perfect numbers are 6, 28, 496, 8128, ... (OEIS A000396), since(3)(4)(5)etc.The th perfect number is implemented in the Wolfram Language as PerfectNumber[n] and checking to see if is a perfect number as PerfectNumberQ[k].The first few perfect numbers are summarized in the following table together with their corresponding indices (see below).1262328354964781285133355033661785898690567191374386913288312305843008139952128Perfect numbers were deemed to have important numerological properties by the ancients, and were extensively studied by the Greeks, including Euclid.Perfect numbers are also intimately connected with a class of numbers known as Mersenne primes, which..

### Lucas number

The Lucas numbers are the sequence of integers defined by the linear recurrence equation(1)with and . The th Lucas number is implemented in the Wolfram Language as LucasL[n].The values of for , 2, ... are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (OEIS A000204).The Lucas numbers are also a Lucas sequence and are the companions to the Fibonacci numbers and satisfy the same recurrence.The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is (Honsberger 1985, p. 122).The only square numbers in the Lucas sequence are 1 and 4 (Alfred 1964, Cohn 1964). The only triangular Lucas numbers are 1, 3, and 5778 (Ming 1991). The only cubic Lucas number is 1.Rather amazingly, if is prime, . The converse does not necessarily hold true, however, and composite numbers such that are known as Lucas pseudoprimes.For , 2, ..., the numbers of decimal digits in are..

### Perrin sequence

The integer sequence defined by the recurrence(1)with the initial conditions , , . This recurrence relation is the same as that for the Padovan sequence but with different initial conditions. The first few terms for , 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608).The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply the Fibonacci numbers). is the solution of a third-order linear homogeneous recurrence equation having characteristic equation(2)Denoting the roots of this equation by , , and , with the unique real root, the solution is then(3)Here,(4)is the plastic constant , which is also given by the limit(5)The asymptotic behavior of is(6)The first few primes in this sequence are 2, 3, 2, 5, 5, 7, 17, 29, 277, 367, 853, ... (OEIS A074788), which occur for terms , 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042,..

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