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A tetradic (or four-way) number is a number that remains unchanged when flipped back to front, mirrored up-down, or flipped up-down. Since the only numbers that remain unchanged which turned up-side-down or mirrored are 0, 1, and 8 (here, the numerals 1 and 8 are assumed to be written as a single stroke and symmetrical pair of loops, respectively), a tetradic number is precisely a palindromic number containing only 0, 1, and 8 as digits. The first few are therefore 1, 8, 11, 88, 101, 111, 181, 808, 818, ... (OEIS A006072).The first few tetradic primes are 11, 101, 181, 18181, 1008001, 1180811, 1880881, 1881881, ... (OEIS A068188). The largest known tetradic prime as of Apr. 2010 iswhere is a repunit (https://primes.utm.edu/top20/page.php?id=53#records), which has decimal digits.

A Smarandache prime is a prime Smarandache number, i.e., a prime number of the form . Surprisingly, no Smarandache primes are known as of Nov. 2015. Upper limits on the non-appearance of primes are summarized in the table below. The search from to was completed by Balatov (2015b), and search of larger terms is now underway (Great Smarandache PRPrime search). As of Dec. 2016, it is known that there are no Smarandache primes up to index 344869.reference200Fleuren (1999)E. Weisstein (Mar. 21, 2009)E. Weisstein (Oct. 17, 2011)M. Alekseyev (Oct. 3, 2015)S. Batalov (Oct. 22 2015)The Great Smarandache PRPrime search (Dec. 5, 2016)If all digit substrings are allowed (so that e.g., concatenating just the 1 from 10, just 10111 from 101112, etc. are permitted), prime digit sequences are known. In particular, such primes are Champernowne-constant primes, the first few of which..

A brute-force method of finding a divisor of an integer by simply plugging in one or a set of integers and seeing if they divide . Repeated application of trial division to obtain the complete prime factorization of a number is called direct search factorization. An individual integer being tested is called a trial divisor.

Fractran is an algorithm applied to a given list , , ..., of fractions. Given a starting integer , the FRACTRAN algorithm proceeds by repeatedly multiplying the integer at a given stage by the first element that yields an integer product. The algorithm terminates when there is no such .The listwith starting integer generates a sequence 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ... (OEIS A007542). Conway (1987) showed that this sequence has an amazing connection with prime numbers, and in fact is a generator for the primes. In particular, the only powers of two (other than 2 itself) that occur in this sequence are those with prime exponent: , , , , ....

Consider the Euler product(1)where is the Riemann zeta function and is the th prime. , but taking the finite product up to , premultiplying by a factor , and letting gives(2)(3)where is the Euler-Mascheroni constant (Havil 2003, p. 173). This amazing result is known as the Mertens theorem.At least for , the sequence of finite products approaches strictly from above (Rosser and Schoenfeld 1962). However, it is highly likely that the finite product is less than its limiting value for infinitely many values of , which is usually the case for any such inequality due to the presence of zeros of on the critical line . An example is Littlewood's famous proof that the sense of the inequality , where is the prime counting function and is the logarithmic integral, reverses infinitely often. While Rosser and Schoenfeld (1962) suggest that "perhaps one can extend [this] result to show that [the Mertens inequality] fails for large ; we have not investigated..

Let be the th Bernoulli number and considerwhere the residues of fractions are taken in the usual way so as to yield integers, for which the minimal residue is taken. Agoh's conjecture states that this quantity is iff is prime. There are no counterexamples less than (S. Plouffe, pers. comm., Jan. 28, 2003). Any counterexample to Agoh's conjecture would be a contradiction to Giuga's conjecture, and vice versa.For , 2, ..., the minimal integer residues (mod ) is 0, , , 0, , 0, , 0, , 0, , ... (OEIS A046094).Kellner (2002) provided a short proof of the equivalence of Giuga'sand Agoh's conjectures. The combined conjecture can be described by a sum of fractions.

The number of ways a set of elements can be partitioned into nonempty subsets is called a Bell number and is denoted (not to be confused with the Bernoulli number, which is also commonly denoted ).For example, there are five ways the numbers can be partitioned: , , , , and , so ., and the first few Bell numbers for , 2, ... are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (OEIS A000110). The numbers of digits in for , 1, ... are given by 1, 6, 116, 1928, 27665, ... (OEIS A113015).Bell numbers are implemented in the WolframLanguage as BellB[n].Though Bell numbers have traditionally been attributed to E. T. Bell as a result of the general theory he developed in his 1934 paper (Bell 1934), the first systematic study of Bell numbers was made by Ramanujan in chapter 3 of his second notebook approximately 25-30 years prior to Bell's work (B. C. Berndt, pers. comm., Jan. 4 and 13, 2010).The first few prime Bell numbers occur at indices..

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..

A Chen prime is a prime number for which is either a prime or semiprime. Chen primes are named after Jing Run Chen who proved in 1966 that there are infinitely many such primes (Chen's theorem).The first Chen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A109611). The first primes that are not Chen primes are 43, 61, 73, 79, 97, 103, 151, ... (OEIS A102540).The lesser of any twin prime is always a Chen prime. Apart from twin prime records, the largest known Chen prime known as of Oct. 2005 was(https://primes.utm.edu/primes/page.php?id=75857),which has 70301 digits.There are infinitely many cases of 3 Chen primes in arithmetic progression (Green and Tao 2005). The following 3074-digit case produces Chen primes for , 1, 2, where denotes the primorial:

Define an emirpimes ("semiprime" spelled backwards) as a semiprime whose (base 10) reversal is a different semiprime. The first such number is 15, because 15 reversed is 51 and both 15 and 51 are semiprimes (i.e., and ). A list of the first emirpimeses (or "semirpimes") are 15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (OEIS A097393). A binary plot of the semirpimes is illustrated above.The numbers of semirpimes less than for , 2, ... are 0, 10, 90, 898, 7200, 60732, ... (OEIS A097394).

A triangle with rows containing the numbers that begins with 1, ends with , and such that the sum of each two consecutive entries being a prime. Rows 2 to 6 are unique,(OEIS A051237) but there are multiple possibilities starting with row 7. For example, the two possibilities for row 7 are and . The number of possible rows ending with , 2, ..., are 0, 1, 1, 1, 1, 1, 2, 4, 7, 24, 80, ... (OEIS A036440).

A Cullen number is a number of the formThe first few are 3, 9, 25, 65, 161, 385, ... (OEIS A002064).Cullen numbers are divisible by if is a prime of the form .The first few prime Cullen numbers are numbers 3, 393050634124102232869567034555427371542904833, ... (OEIS A050920), corresponding to , 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881, ... (OEIS A005849; Caldwell). As of Nov. 2015, it is known that there are no other Cullen primes for (PrimeGrid).

The tribonacci numbers are a generalization of the Fibonacci numbers defined by , , , and the recurrence equation(1)for (e.g., Develin 2000). They represent the case of the Fibonacci n-step numbers.The first few terms using the above indexing convention for , 1, 2, ... are 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (OEIS A000073; which however adopts the alternate indexing convention and ).The first few prime tribonacci numbers are 2, 7, 13, 149, 19341322569415713958901, ... (OEIS A092836), which have indices 3, 5, 6, 10, 86, 97, 214, 801, 4201, 18698, 96878, ... (OEIS A092835), and no others with (E. W. Weisstein, Mar. 21, 2009).Using Brown's criterion, it can be shown that the tribonacci numbers are complete; that is, every positive number can be written as the sum of distinct tribonacci numbers. Moreover, every positive number has a unique Zeckendorf-like expansion as the sum of distinct tribonacci numbers and that sum does..

A prime for which has a maximal period decimal expansion of digits. Full reptend primes are sometimes also called long primes (Conway and Guy 1996, pp. 157-163 and 166-171). There is a surprising connection between full reptend primes and Fermat primes.A prime is full reptend iff 10 is a primitive root modulo , which means that(1)for and no less than this. In other words, the multiplicative order of (mod 10) is . For example, 7 is a full reptend prime since .The full reptend primes are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ... (OEIS A001913). The first few decimal expansions of these are(2)(3)(4)(5)Here, the numbers 142857, 5882352941176470, 526315789473684210, ... (OEIS A004042) corresponding to the periodic parts of these decimal expansions are called cyclic numbers. No general method is known for finding full reptend primes.The number of full reptend primes less than for , 2, ... are 1, 9, 60, 467, 3617, ... (OEIS A086018).A..

A prime circle of order is a free circular permutation of the numbers from 1 to with adjacent pairs summing to a prime. The number of prime circles for , 2, ..., are 1, 1, 1, 2, 48, 512, ... (OEIS A051252). The prime circles for the first few even orders are given in the table below.prime circles2468,

A prime partition of a positive integer is a set of primes which sum to . For example, there are three prime partitions of 7 sinceThe number of prime partitions of , 3, ... are 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, ... (OEIS A000607). If for prime and for composite, then the Euler transform gives the number of partitions of into prime parts (Sloane and Plouffe 1995, p. 21).The minimum number of primes needed to sum to , 3, ... are 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, ... (OEIS A051034). The maximum number of primes needed to sum to is just , 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... (OEIS A004526), corresponding to a representation in terms of all 2s for an even number or one 3 and the rest 2s for an odd number.The numbers which can be represented by a single prime are obviously the primes themselves. Composite numbers which can be represented as the sum of two primes are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ... (OEIS A051035), and composite..

The fibonorial , also called the Fibonacci factorial, is defined aswhere is a Fibonacci number. For , 2, ..., the first few fibonorials are 1, 1, 2, 6, 30, 240, 3120, 65520, ... (OEIS A003266).The fibonorials are asymptotic towhere is the Fibonacci factorial constant and is the golden ratio.The first few values of such that is prime are given by 4, 5, 6, 7, 8, 14, 15, ... (OEIS A059709), with no others less than 500.The first few values of such that is prime are given by 1, 2, 3, 4, 5, 6, 7, 8, 22, 28, ... (OEIS A053408), with no others less than 500.

Given a positive integer , let its prime factorization be written(1)Define the functions and by , , and(2)(3)The first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, ... (OEIS A051904), while the first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, ... (OEIS A051903).Then the average value of tends to(4)Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 16/13, ... (OEIS A086195 and A086196).In addition, the ratio(5)where is the Riemann zeta function (Niven 1969).Niven (1969) also proved that(6)where Niven's constant is given by(7)(OEIS A033150). Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 17/13, ... (OEIS A086197 and A086198).The continued fraction of Niven's constant is 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, ... (OEIS A033151). The positions at which the digits 1, 2, ... first occur in the continued fraction are 1, 3, 10, 7, 47, 41, 34,..

Mills (1947) proved the existence of a real constant such that(1)is prime for all integers , where is the floor function. Mills (1947) did not, however, determine , or even a range for .A generalization of Mills' theorem to an arbitrary sequence of positiveintegers is given as an exercise by Ellison and Ellison (1985).The least such that is prime for all integers is known as Mills' constant.Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let be the th prime, then there exists a constant such that(2)for all . This has more recently been strengthened to(3)(Mozzochi 1986). If the Riemann hypothesisis true, then Cramér (1937) showed that(4)(Finch 2003).Hardy and Wright (1979) and Ribenboim (1996) point out that, despite the beauty of such prime formulas, they do not have any practical consequences. In fact, unless the exact value of is known, the primes themselves must be known in advance to determine..

Mills' theorem states that there exists a real constant such that is prime for all positive integers (Mills 1947). While for each value of , there are uncountably many possible values of such that is prime for all positive integers (Caldwell and Cheng 2005), it is possible to define Mills' constant as the least such thatis prime for all positive integers , giving a value of(OEIS A051021). is therefore given by the next prime after , and the values of are known as Mills' primes (Caldwell and Cheng 2005).Caldwell and Cheng (2005) computed more than 6850 digits of assuming the truth of the Riemann hypothesis. Proof of primality of the 13 Mills prime in Jul. 2013 means that approximately digits are now known.It is not known if is irrational.

The Mertens constant , also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant (Bach and Shallit 1996, p. 234), or Kronecker's constant (Schroeder 1997), is a constant related to the twin primes constant and that appears in Mertens' second theorem,(1)where the sum is over primes and is a Landau symbol. This sum is the analog of(2)where is the Euler-Mascheroni constant (Gourdon and Sebah).The constant is given by the infinite sum(3)where is the Euler-Mascheroni constant and is the th prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979; Le Lionnais 1983; Ellison and Ellison 1985), or by the limit(4)According to Lindqvist and Peetre (1997), this was shown independently by Meisselin 1866 and Mertens (1874). Formula (3) is equivalent to(5)which follows from (4) using the Mercator series for with . is also given by the rapidly converging series(6)where is the Riemann zeta function, and is the Möbius..

Let be the smallest prime in the arithmetic progression for an integer . Letsuch that and . Then there exists a and an such that for all . is known as Linnik's constant.

An algorithm for making tables of primes. Sequentially write down the integers from 2 to the highest number you wish to include in the table. Cross out all numbers which are divisible by 2 (every second number). Find the smallest remaining number . It is 3. So cross out all numbers which are divisible by 3 (every third number). Find the smallest remaining number . It is 5. So cross out all numbers which are divisible by 5 (every fifth number).Continue until you have crossed out all numbers divisible by , where is the floor function. The numbers remaining are prime. This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out composite numbers up to . If the procedure is then continued up to , then the number of cross-outs gives the number of distinct prime factors of each number.The sieve of Eratosthenes can be used to compute the primecounting function aswhich is essentially an application of the inclusion-exclusionprinciple..

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers . Pick values of given by(1)where , 2, ... and is the floor function. We are then looking for factors such that(2)which means that only numbers with Legendre symbol (less than for trial divisor , where is the prime counting function) need be considered. The set of primes for which this is true is known as the factor base. Next, the congruences(3)must be solved for each in the factor base. Finally, a sieve is applied to find values of which can be factored completely using only the factor base. Gaussian elimination is then used as in Dixon's factorization method in order to find a product of the s, yielding a perfect square.The method requires about steps, improving on the continued fraction factorization algorithm by removing the 2 under the square root (Pomerance 1996). The use of multiple polynomials gives a better chance of factorization,..

The prime spiral, also known as Ulam's spiral, is a plot in which the positive integers are arranged in a spiral (left figure), with primes indicated in some way along the spiral. In the right plot above, primes are indicated in red and composites are indicated in yellow.The plot above shows a larger part of the spiral in which the primes are shown as dots.Unexpected patterns of diagonal lines are apparent in such a plot, as illustrated in the above grid. This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly..

Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that be a prime is that be a prime of one of the forms , , ."Mersenne's implication has been refuted, but Bateman, Selfridge, and Wagstaff (1989) used the statement as an inspiration for what is now called the new Mersenne conjecture, which can be stated as follows.Consider an odd natural number . If two of the following conditions hold, then so does the third: 1. or , 2. is prime (a Mersenne prime), 3. is prime (a Wagstaff prime). This conjecture has been verified for all primes .Based on the distribution and heuristics of (cf. https://www.utm.edu/research/primes/mersenne/heuristic.html) the known Mersenne and Wagstaff prime exponents, it seems quite likely that there is only a finite number of exponents satisfying the criteria of the new Mersenne conjecture. In..

Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely: 1. The Goldbach conjecture, 2. Twin prime conjecture, 3. Legendre's conjecture that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and 4. The conjecture that there are infinitely many primes of the form (Euler 1760; Mirsky 1949; Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208). The first few such primes are 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496). Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest primes between and for , 2, ..., are 2, 5, 11,..

A prime magic square is a magic square consisting only of prime numbers (although the number 1 is sometimes allowed in such squares). The left square is the prime magic square (containing a 1) having the smallest possible magic constant, and was discovered by Dudeney in 1917 (Dudeney 1970; Gardner 1984, p. 86). The second square is the magic square consisting of primes only having the smallest possible magic constant (Madachy 1979, p. 95; attributed to R. Ondrejka). The third square is the prime magic square consisting of primes in arithmetic progression () having the smallest possible magic constant of 3117 (Madachy 1979, p. 95; attributed to R. Ondrejka). The prime magic square on the right was found by A. W. Johnson, Jr. (Dewdney 1988).According to a 1913 proof of J. N. Muncey (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive odd primes including..

If divides the numerator of the Bernoulli number for , then is called an irregular pair. For , the irregular pairs of various forms are for , for , none for , and for .

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

, 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,..

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,..

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..

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..

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..

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)...

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)

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..

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 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..

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 ...

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.

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...

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.

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..

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..

A generalization of Fermat's last theorem which states that if , where , , , , , and are any positive integers with , then , , and have a common factor. The conjecture was announced in Mauldin (1997), and a cash prize of has been offered for its proof or a counterexample (Castelvecchi 2013).This conjecture is more properly known as the Tijdeman-Zagier conjecture (Elkies 2007).

Like the entire harmonic series, the harmonicseries(1)taken over all primes also diverges, as first shown by Euler in 1737 (Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17 and 22; Wells 1986, p. 41; Havil 2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (OEIS A046024) primes.Its asymptotic behavior is given by(2)where(3)(OEIS A077761) is the Mertens constant (Hardy and Wright 1979, p. 351; Hardy 1999, p. 50; Havil 2003, p. 64).

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..

The converse of Fermat's little theorem is also known as Lehmer's theorem. It states that, if an integer is prime to and and there is no integer for which , then is not prime. Here, is called a witness to the primality of . This theorem is the basis for the Pratt primality certificate.

Define a Bouniakowsky polynomial as an irreducible polynomial with integer coefficients, degree , and . The Bouniakowsky conjecture states that is prime for an infinite number of integers (Bouniakowsky 1857). As an example of the greatest common divisor caveat, the polynomial is irreducible, but always divisible by 2.Irreducible degree 1 polynomials () always generate an infinite number of primes by Dirichlet's theorem. The existence of a Bouniakowsky polynomial that can produce an infinitude of primes is undetermined. The weaker fifth Hardy-Littlewood conjecture asserts that is prime for an infinite number of integers .Various prime-generating polynomialsare known, but none of these always generates a prime (Legendre).Worse yet, it is unknown if a general Bouniakowsky polynomial will always produce at least 1 prime number. For example, produces no primes until , 764400, 933660, ... (OEIS A122131)...

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..

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)

The prime zeta function(1)where the sum is taken over primes is a generalizationof the Riemann zeta function(2)where the sum is over all positive integers. In other words, the prime zeta function is the Dirichlet generating function of the characteristic function of the primes . is illustrated above on positive the real axis, where the imaginary part is indicated in yellow and the real part in red. (The sign difference in the imaginary part compared to the plot appearing in Fröberg is presumably a result of the use of a different convention for .)Various terms and notations are used for this function. The term "prime zeta function" and notation were used by Fröberg (1968), whereas Cohen (2000) uses the notation .The series converges absolutely for , where , can be analytically continued to the strip (Fröberg 1968), but not beyond the line (Landau and Walfisz 1920, Fröberg 1968) due to the clustering of singular..

Honaker's problem asks for all consecutive prime number triples with such that . Caldwell and Cheng (2005) showed that the only Honaker triplets for are (2, 3, 5), (3, 5, 7), and (61, 67, 71). In addition, Caldwell and Cheng (2005) showed that the Cramér-Granville conjecture implies that there can only exist a finite number of such triplets, that implies there are exactly three, and conjectured that these three are in fact the only such triplets.

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..

The integer sequence defined by the recurrencerelation(1)with the initial conditions . This is the same recurrence relation as for the Perrin sequence, but with different initial conditions.The recurrence relation can be solved explicitly,giving(2)where is the th root of(3)Another form of the solution is(4)where is the th root of(5)The first few terms are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, ... (OEIS A000931).The first few prime Padovan numbers are 2, 2, 3, 5, 7, 37, 151, 3329, 23833, ... (OEIS A100891), corresponding to indices ,3, 4, 5, 7, 8, 14, 19, 30, 37, 84, 128, 469, 666, 1262, 1573, 2003, 2210, 2289, 4163, 5553, 6567, 8561, 11230, 18737, 35834, 44259, 536485, ... (OEIS A112882). The search for prime numerators has been completed up to by E. W. Weisstein (Apr. 10, 2011), and the following table summarizes the largest known values.decimal digitsdiscoverer53648565518E. W. Weisstein (May 16, 2009)72773488874E. W. Weisstein..

While the Catalan numbers are the number of p-good paths from to (0,0) which do not cross the diagonal line, the super Catalan numbers count the number of lattice paths with diagonal steps from to (0,0) which do not touch the diagonal line .The super Catalan numbers are given by the recurrencerelation(1)(Comtet 1974), with . (Note that the expression in Vardi (1991, p. 198) contains two errors.) A closed form expression in terms of Legendre polynomials for is(2)(3)(Vardi 1991, p. 199). The first few super Catalan numbers are 1, 1, 3, 11, 45, 197, ... (OEIS A001003). These are often called the "little" Schröder numbers. Multiplying by 2 gives the usual ("large") Schröder numbers 2, 6, 22, 90, ... (OEIS A006318).The first few prime super Catalan numbers have indices 3, 4, 6, 10, 216, ... (OEIS A092839), with no others less than (Weisstein, Mar. 7, 2004), corresponding to the numbers 3, 11, 197,..

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..

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,..

Apéry's numbers are defined by(1)(2)(3)where is a binomial coefficient. The first few for , 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).The first few prime Apéry numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092826), which have indices , 2, 12, 24, ... (OEIS A092825).The case of Schmidt's problem expresses these numbers in the form(4)(Strehl 1993, 1994; Koepf 1998, p. 55).They are also given by the recurrence equation(5)with and (Beukers 1987).There is also an associated set of numbers(6)(7)(Beukers 1987), where is a generalized hypergeometric function. The values for , 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS A005258). The first few prime -numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092827), which have indices , 2, 6, 8, ... (OEIS A092828), with no others for (Weisstein, Mar. 8, 2004).The..

The Pell numbers are the numbers obtained by the s in the Lucas sequence with and . They correspond to the Pell polynomial . Similarly, the Pell-Lucas numbers are the s in the Lucas sequence with and , and correspond to the Pell-Lucas polynomial .The Pell numbers and Pell-Lucas numbers are also equal to(1)(2)where is a Fibonacci polynomial.The Pell and Pell-Lucas numbers satisfy the recurrencerelation(3)with initial conditions and for the Pell numbers and for the Pell-Lucas numbers.The th Pell and Pell-Lucas numbers are explicitly given by the Binet-type formulas(4)(5)The th Pell and Pell-Lucas numbers are given by the binomial sums(6)(7)respectively.The Pell and Pell-Lucas numbers satisfy the identities(8)(9)(10)and(11)(12)For , 1, ..., the Pell numbers are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (OEIS A000129).For a Pell number to be prime, it is necessary that be prime. The indices of (probable) prime Pell numbers are 2, 3, 5,..

Primorial primes are primes of the form , where is the primorial of . A coordinated search for such primes is being conducted on PrimeGrid. is prime for , 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... (OEIS A057704; Guy 1994, pp. 7-8; Caldwell 1995). These correspond to with , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, ... (OEIS A006794). The largest known primorial primes as of Nov. 2015 are summarized in the following table (Caldwell).digitsdiscoverer6845Dec. 1992365851PrimeGrid (Dec. 20, 2010)476311PrimeGrid (Mar. 5, 2012) (also known as a Euclid number) is prime for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... (OEIS A014545; Guy 1994, Caldwell 1995, Mudge 1997). These correspond to with , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547,..

Let be the th prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by(1)The values of for , 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (OEIS A002110).It is sometimes convenient to define the primorial for values other than just the primes, in which case it is taken to be given by the product of all primes less than or equal to , i.e.,(2)where is the prime counting function. For , 2, ..., the first few values of are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS A034386).The logarithm of is closely related to the Chebyshev function , and a trivial rearrangement of the limit(3)gives(4)(Ruiz 1997; Finch 2003, p. 14; Pruitt), where eis the usual base of the natural logarithm.

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