A fraction containing each of the digits 1 through 9 is called a pandigital fraction. The following table gives the number of pandigital fractions which represent simple unit fractions. The numbers of pandigital fractions for 1/1, 1/2, 1/3, ... are 0, 12, 2, 4, 12, 3, 7, 46, 3, ... (OEIS A054383).#fractions12,2412,37,46,,,,,,,3004
A Sierpiński number of the second kind is a number satisfying Sierpiński's composite number theorem, i.e., a Proth number such that is composite for every .The smallest known example is , proved in 1962 by J. Selfridge, but the fate of a number of smaller candidates remains to be determined before this number can be established as the smallest such number. As of 1996, 35 candidates remained (Ribenboim 1996, p. 358), a number which had been reduced to 17 by the beginning of 2002 (Peterson 2003).In March 2002, L. K. Helm and D. A. Norris began a distributed computing effort dubbed "seventeen or bust" to eliminate the remaining candidates. With the aid of collaborators across the globe, this number was reduced to 12 as of December 2003 (Peterson 2003, Helm and Norris). The following table summarizes numbers subsequently found to be prime by "seventeen or bust," leaving only..
It is thought that the totient valence function , i.e., if there is an such that , then there are at least two solutions . This assertion is called Carmichael's totient function conjecture and is equivalent to the statement that there exists an such that (Ribenboim 1996, pp. 39-40).Dickson (2005, p. 137) states that the conjecture was proved by Carmichael (1907), who also developed a method of finding the solution (Carmichael 1909). The result also appears as in exercise in Carmichael (1914). However, Carmichael (1922) subsequently discovered an error in the proof, and the conjecture currently remains open. Any counterexample to the conjecture must have more than digits (Schlafly and Wagon 1994; conservatively given as in Conway and Guy 1996, p. 155). This result was extended by Ford (1999), who showed that any counterexample must have more than digits.Ford (1998ab) showed that if there is a counterexample to Carmichael's..
The central binomial coefficient is never squarefree for . This was proved true for all sufficiently large by Sárkőzy's theorem. Goetgheluck (1988) proved the conjecture true for and Vardi (1991) for . The conjecture was proved true in its entirety by Granville and Ramare (1996).
The complexity of an integer is the least number of 1s needed to represent it using only additions, multiplications, and parentheses. For example, the numbers 1 through 10 can be minimally represented as(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)so the complexities for , 2, ..., are 1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, ... (OEIS A005245).The smallest numbers of complexity , 2, ... are 1, 2, 3, 4, 5, 7, 10, 11, 17, 22, 23, 41, ... (OEIS A005520).
The weak law of large numbers (cf. the strong law of large numbers) is a result in probability theory also known as Bernoulli's theorem. Let , ..., be a sequence of independent and identically distributed random variables, each having a mean and standard deviation . Define a new variable(1)Then, as , the sample mean equals the population mean of each variable.(2)(3)(4)(5)In addition,(6)(7)(8)(9)Therefore, by the Chebyshev inequality, for all ,(10)As , it then follows that(11)(Khinchin 1929). Stated another way, the probability that the average for an arbitrary positive quantity approaches 1 as (Feller 1968, pp. 228-229).
A Sierpiński number of the first kind is a number of the form . The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (OEIS A014566). Sierpiński proved that if is prime with , then must be of the form , making a Fermat number with . The first few of this form are 1, 3, 6, 11, 20, 37, 70, ... (OEIS A006127).The numbers of digits in the number is given bywhere is the ceiling function, so the numbers of digits in the first few candidates are 1, 3, 20, 617, 315653, 41373247568, ... (OEIS A089943).The only known prime Sierpiński numbers of the first kind are 2, 5, 257, with the first unknown case being . The status of Sierpiński numbers is summarized in the table below (Nielsen).status of 01prime ()13prime ()26composite with factor 311composite with factor 420composite with no factor known537composite with factor 670unknown7135unknown8264unknown9521unknown101034unknown112059composite with factor 124108unknown138205unknown1416398unknown1532783unknown1665552unknown17131089unknown..
A Mersenne number is a number of the form(1)where is an integer. The Mersenne numbers consist of all 1s in base-2, and are therefore binary repunits. The first few Mersenne numbers are 1, 3, 7, 15, 31, 63, 127, 255, ... (OEIS A000225), corresponding to , , , , ... in binary.The Mersenne numbers are also the numbers obtained by setting in a Fermat polynomial. They also correspond to Cunningham numbers .The number of digits in the Mersenne number is(2)where is the floor function, which, for large , gives(3)The number of digits in is the same as the number of digits in , namely 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, ... (OEIS A034887). The numbers of decimal digits in for , 1, ... are given by 1, 4, 31, 302, 3011, 30103, 301030, 3010300, 30103000, 301029996, ... (OEIS A114475), which correspond to the decimal expansion of (OEIS A007524).The numbers of prime factors of for , 2, ... are 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, ... (OEIS A046051), and the first few..
A Colbert number is any prime number with more than decimal digits whose discovery contributes to the long-sought after proof that is the smallest Sierpiński number of the second kind. Colbert Numbers are named to honor Stephen T. Colbert.There are currently five known Colbert numbers, as summarized in the following table.Colbert numberdecimal digits15215613918990275967723572072116617The Seventeen or Bust distributed computing effort is conducting a search for the remaining six Colbert numbers (where indicates the exponent is unknown).unknown Colbert numberdecimal digits??????
The prime number theorem shows that the th prime number has the asymptotic value(1)as (Havil 2003, p. 182). Rosser's theorem makes this a rigorous lower bound by stating that(2)for (Rosser 1938). This result was subsequently improved to(3)where (Rosser and Schoenfeld 1975). The constant was subsequently reduced to (Robin 1983). Massias and Robin (1996) then showed that was admissible for and . Finally, Dusart (1999) showed that holds for all (Havil 2003, p. 183). The plots above show (black), (blue), and (red).The difference between and is plotted above. The slope of the difference taken out to is approximately .
The Skewes number (or first Skewes number) is the number above which must fail (assuming that the Riemann hypothesis is true), where is the prime counting function and is the logarithmic integral.Isaac Asimov featured the Skewes number in his science fact article "Skewered!"(1974).In 1912, Littlewood proved that exists (Hardy 1999, p. 17), and the upper boundwas subsequently found by Skewes (1933). The Skewes number has since been reduced to by Lehman in 1966 (Conway and Guy 1996; Derbyshire 2004, p. 237), by te Riele (1987), and less than (Bays and Hudson 2000; Granville 2002; Borwein and Bailey 2003, p. 65; Havil 2003, p. 200; Derbyshire 2004, p. 237). The results of Bays and Hudson left open the possibility that the inequality could fail around , and thus established a large range of violation around (Derbyshire 2004, p. 237). More recent work by Demichel establishes that the first crossover..
Wang's conjecture states that if a set of tiles can tile the plane, then they can always be arranged to do so periodically (Wang 1961). The conjecture was refuted when Berger (1966) showed that an aperiodic set of tiles existed. Berger used tiles, but the number has subsequently been greatly reduced. In fact, Culik (1996) has reduced the number of colored square tiles to 13.For purely square tiles, Culik's record still stands as of Feb. 2009. For non-square tiles, it is much more complicated due to the Penrose tiles (2 tiles), the Robertson tiling (6 tiles), and various Ammann tilings (2-5 tiles).
A number having 666 digits (where 666 is the beastnumber) is called an apocalypse number.The Fibonacci number is the smallest Fibonacci apocalypse number (Livio 2002, p. 108).Apocalypse primes are given by for , 1837, 6409, 7329, 8569, 8967, 9663, ... (OEIS A115983). The smallest apocalypse prime containing the digits 666 is (Rupinski).
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..
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.
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..
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..
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 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.
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..
Let be the smallest dimension of a hypercube such that if the lines joining all pairs of corners are two-colored for any , a complete graph of one color with coplanar vertices will be forced. Stated colloquially, this definition is equivalent to considering every possible committee from some number of people and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find the smallest that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees (Hoffman 1998, p. 54).An answer was proved to exist by Graham and Rothschild (1971), who also provided the best known upper bound, given by(1)where Graham's number is recursively defined by(2)and(3)Here, is the so-called Knuth up-arrow notation. is often cited as the largest number that has ever been put to practical use (Exoo 2003).In chained arrow notation, satisfies..
The number , where 666 is the beast number and denotes a factorial. The number has approximately decimal digits.The number of trailing zeros in the Leviathan number is(1)(2)(Pickover 1995).
Legion's number of the first kind is defined as(1)(2)where 666 is the beast number. It has 1881 decimaldigits.Legion's number of the second kind is defined as(3)(4)It has approximately digits, and ends with trailing zeros.
The number of digits in an integer is the number of numbers in some base (usually 10) required to represent it. The numbers 1 to 9 are therefore single digits, while the numbers 10 to 99 are double digits. Terms such as "double-digit inflation" are occasionally encountered, although this particular usage has thankfully not been needed in the U.S. for some time. The number of base- digits in a number can be calculated as(1)where is the floor function. For , the formula becomes(2)The number of digits in the number represented in base is given by the Wolfram Language function DigitCount[n, b, d], with DigitCount[n, b] giving a list of the numbers of each digit in . The total number of digits in a number is given by IntegerLength[n, b].The positive integers with distinct base-10 digits are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ... (OEIS A010784). The number of -digit integers is given by(3)(4)(5)(6)where is..