1.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers

2.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n

3.
Primorial
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The rest of this article uses the latter interpretation. The name primorial, coined by Harvey Dubner, draws an analogy to primes the same way the name relates to factors. For the nth prime number pn, the primorial pn# is defined as the product of the first n primes, p n # ≡ ∏ k =1 n p k, where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes, the first six primorials pn# are,1,2,6,30,210,2310. The sequence also includes p0# =1 as empty product, asymptotically, primorials pn# grow according to, p n # = e n log n, where o is little-o notation. This is equivalent to, n # = {1 if n =0,1 # × n if n is prime # if n is composite. For example, 12# represents the product of those primes ≤12,12 # =2 ×3 ×5 ×7 ×11 =2310, since π =5, this can be calculated as,12 # = p π # = p 5 # =2310. Consider the first 12 primorials n#,1,2,6,6,30,30,210,210,210,210,2310,2310. We see that for composite n every term n# simply duplicates the preceding term #, in the above example we have 12# = p5# = 11# since 12 is a composite number. The natural logarithm of n# is the first Chebyshev function, written ϑ or θ, primorials n# grow according to, ln ∼ n. The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 7009223613394100000♠2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes. Every highly composite number is a product of primorials, primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction φ/n is smaller than it for any lesser integer, any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values. Base systems corresponding to primorials have a proportion of repeating fractions than any smaller base. Every primorial is a sparsely totient number, the n-compositorial of a composite number n is the product of all composite numbers up to and including n. The n-compositorial is equal to the n-factorial divided by the primorial n#, the compositorials are 1,4,24,192,1728, 7004172800000000000♠17280, 7005207360000000000♠207360, 7006290304000000000♠2903040, 7007435456000000000♠43545600, 7008696729600000000♠696729600

4.
Partition (number theory)
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In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition, a summand in a partition is also called a part. The number of partitions of n is given by the function p. The notation λ ⊢ n means that λ is a partition of n, Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general. For example, the partition 2 +2 +1 might instead be written as the tuple or in the more compact form where the superscript indicates the number of repetitions of a term. There are two common methods to represent partitions, as Ferrers diagrams, named after Norman Macleod Ferrers. Both have several possible conventions, here, we use English notation, with diagrams aligned in the upper-left corner. The partition 6 +4 +3 +1 of the positive number 14 can be represented by the diagram, The 14 circles are lined up in 4 rows. The diagrams for the 5 partitions of the number 4 are listed below, rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. As a type of shape made by adjacent squares joined together, by convention p =1, p =0 for n negative. The first few values of the function are,1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,1575,1958,2436,3010,3718,4565,5604. As of June 2013, the largest known prime number that counts a number of partitions is p, the generating function for p is given by, ∑ n =0 ∞ p x n = ∏ k =1 ∞. Expanding each factor on the side as a geometric series. The xn term in this product counts the number of ways to write n = a1 + 2a2 + 3a3 +, where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the generating function. More generally, the function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A. This result is due to Euler, the formulation of Eulers generating function is a special case of a q-Pochhammer symbol and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function

5.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers

6.
PrimeGrid
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PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform, PrimeGrid started in June 2005 under the name Message@home and tried to decipher text fragments hashed with MD5. Message@home was a test to port the BOINC scheduler to Perl to obtain greater portability, after a while the project attempted the RSA factoring challenge trying to factor RSA-640. After RSA-640 was factored by a team in November 2005. With the chance to succeed too small, it discarded the RSA challenges, was renamed to PrimeGrid, at 210,000,000,000 the primegen subproject was stopped. In June 2006, dialog started with Riesel Sieve to bring their project to the BOINC community, PrimeGrid provided PerlBOINC support and Riesel Sieve was successful in implementing their sieve as well as a prime finding application. With collaboration from Riesel Sieve, PrimeGrid was able to implement the LLR application in partnership with another prime finding project, in November 2006, the TPS LLR application was officially released at PrimeGrid. Less than two months later, January 2007, the twin was found by the original manual project. PrimeGrid and TPS then advanced their search for even larger twin primes, the summer of 2007 was very active as the Cullen and Woodall prime searches were launched. In the Fall, more prime searches were added through partnerships with the Prime Sierpinski Problem, additionally, two sieves were added, the Prime Sierpinski Problem combined sieve which includes supporting the Seventeen or Bust sieve, and the combined Cullen/Woodall sieve. In the Fall of 2007, PrimeGrid migrated its systems from PerlBOINC to standard BOINC software, since September 2008, PrimeGrid is also running a Proth prime sieving subproject. In January 2010 the subproject Seventeen or Bust was added, the calculations for the Riesel problem followed in March 2010. In addition, PrimeGrid is helping test for a record Sophie Germain prime. As of March 2016, PrimeGrid is working on or has worked on the projects,321 Prime Search is a continuation of Paul Underwoods 321 Search which looked for primes of the form 3 · 2n −1. PrimeGrid added the +1 form and continues the search up to n = 25M, the search was successful in April 2010 with the finding of the first known AP26,43142746595714191 +23681770 · 23# · n is prime for n =0. 23# = 2·3·5·7·11·13·17·19·23 =223092870, or 23 primorial, is the product of all primes up to 23, PrimeGrid is also running a search for Cullen prime numbers, yielding the two largest known Cullen primes. The first one being the 14th largest known prime at the time of discovery, as of 9 March 2014 PrimeGrid has eliminated 14 values of k from the Riesel problem and is continuing the search to eliminate the 50 remaining numbers. Primegrid then worked with the Twin Prime Search to search for a twin prime at approximately 58700 digits

7.
Primecoin
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Primecoin is a peer-to-peer open source cryptocurrency that implements a unique scientific computing proof-of-work system. Primecoins proof-of-work system searches for chains of prime numbers, Primecoin was created by a person or group of people who use the pseudonym Sunny King. This entity is also related with the cryptocurrency Peercoin, the Primecoin source code is copyrighted by a person or group called “Primecoin Developers”, and distributed under a conditional MIT/X11 software license. Primecoin has been described as the cause of spot shortages of dedicated servers because at the time it was only possible to mine the currency with CPUs. For the same reason, Primecoin used to be the target of malware writers, there has been some attempts at approximating this number. Difficulty adjustment is more frequent Primecoin protocol adjusts its difficulty slightly after every block, the difficulty change that occurs each block is targeted at achieving target of one new block created once per minute. As comparison the bitcoin protocol adjusts its difficulty every 2016 blocks, faster transaction confirmations Since Primecoin blocks are generated 8 to 10 times as fast as bitcoin blocks on average, Primecoin transactions are confirmed approximately 8 to 10 times as fast. Primecoin uses the finding of prime chains composed of Cunningham chains and bi-twin chains for proof-of-work, the system is designed so that the work is efficiently verifiable by all nodes on the Primecoin network. To meet this requirement, the size of the numbers in the system cannot be too large. The Primecoin proof-of-work system has the characteristics, Primecoins work takes the form of prime number chains. Finding the prime number chains becomes exponentially harder as the length is increased. Verification of the reasonably sized prime number chains can be performed efficiently by all network nodes, mersenne primes are precluded due to their extremely large size. Three types of prime number chains are accepted as proof-of-work, Cunningham chain of the first kind, Cunningham chain of the second kind. Other cryptocurrencies including bitcoin commonly use a Hashcash type of proof-of-work based on SHA-256 hash calculations, list of largest known Cunningham chains of given length includes several results generated by Primecoin miners. Anarcho-capitalism Anonymous Internet banking Alternative currency Crypto-anarchism Money supply Moores law Prime number Cunningham chain Official website

8.
Pierpont prime
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A Pierpont prime is a prime number of the form 2 u 3 v +1 for some nonnegative integers u and v. That is, they are the prime numbers p for which p −1 is 3-smooth. They are named after the mathematician James Pierpont, who introduced them in the study of regular polygons that can be constructed using conic sections. It is possible to prove that if v =0 and u >0, then u must be a power of 2, if v is positive then u must also be positive, and the Pierpont prime is of the form 6k +1. Empirically, the Pierpont primes do not seem to be rare or sparsely distributed. There are 36 Pierpont primes less than 106,59 less than 109,151 less than 1020, there are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. As there are Θ numbers of the form in this range. Andrew M. Gleason made this explicit, conjecturing there are infinitely many Pierpont primes. According to Gleasons conjecture there are Θ Pierpont primes smaller than N, when 2 u >3 v, the primality of 2 u 3 v +1 can be tested by Proths theorem. As part of the ongoing search for factors of Fermat numbers. The following table gives values of m, k, and n such that k ⋅2 n +1 divides 22 m +1, the left-hand side is a Pierpont prime when k is a power of 3, the right-hand side is a Fermat number. As of 2017, the largest known Pierpont prime is 3 ×210829346 +1, whose primality was discovered by Sai Yik Tang, in the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation. It follows that they allow any regular polygon of N sides to be formed, as long as N >3 and of the form 2m3nρ and this is the same class of regular polygons as those that can be constructed with a compass, straightedge, and angle-trisector. Regular polygons which can be constructed with compass and straightedge are the special case where n =0 and ρ is a product of distinct Fermat primes, themselves a subset of Pierpont primes. In 1895, James Pierpont studied the same class of regular polygons, Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from previously constructed points. As he showed, the regular N-gons that can be constructed with these operations are the ones such that the totient of N is 3-smooth. Since the totient of a prime is formed by subtracting one from it, however, Pierpont did not describe the form of the composite numbers with 3-smooth totients. As Gleason later showed, these numbers are exactly the ones of the form 2m3nρ given above, the smallest prime that is not a Pierpont prime is 11, therefore, the hendecagon is the smallest regular polygon that cannot be constructed with compass, straightedge and angle trisector

9.
Motzkin number
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In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords between n points on a circle. The Motzkin numbers are named after Theodore Motzkin, and have diverse applications in geometry, combinatorics. The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle, the following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle. Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers, a Motzkin prime is a Motzkin number that is prime. Guibert, Pergola & Pinzani showed that vexillary involutions are enumerated by Motzkin numbers

10.
Pythagorean prime
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A Pythagorean prime is a prime number of the form 4n +1. Pythagorean primes are exactly the odd numbers that are the sum of two squares. For instance, the number 5 is a Pythagorean prime, √5 is the hypotenuse of a triangle with legs 1 and 2. The first few Pythagorean primes are 5,13,17,29,37,41,53,61,73,89,97,101,109,113, by Dirichlets theorem on arithmetic progressions, this sequence is infinite. More strongly, for n, the numbers of Pythagorean and non-Pythagorean primes up to n are approximately equal. However, the number of Pythagorean primes up to n is frequently smaller than the number of non-Pythagorean primes. For example, the values of n up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes are 26861 and 26862. Sum of one odd square and one square is congruent to 1 mod 4. Fermats theorem on sums of two states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4. The representation of such number is unique, up to the ordering of the two squares. Another way to understand this representation as a sum of two squares involves Gaussian integers, the numbers whose real part and imaginary part are both integers. The norm of a Gaussian integer x + yi is the number x2 + y2, thus, the Pythagorean primes occur as norms of Gaussian integers, while other primes do not. Within the Gaussian integers, the Pythagorean primes are not considered to be prime numbers, similarly, their squares can be factored in a different way than their integer factorization, as p2 =22 =. The real and imaginary parts of the factors in these factorizations are the leg lengths of the right triangles having the given hypotenuses, in the finite field Z/p with p a Pythagorean prime, the polynomial equation x2 = −1 has two solutions. This may be expressed by saying that −1 is a quadratic residue mod p, in contrast, this equation has no solution in the finite fields Z/p where p is an odd prime but is not Pythagorean. Pythagorean Primes, including 5,13 and 137, sloanes A007350, Where prime race 4n-1 vs. 4n+1 changes leader. The On-Line Encyclopedia of Integer Sequences

11.
Bell number
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In combinatorial mathematics, the Bell numbers count the number of partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. The nth of these numbers, Bn, counts the number of different ways to partition a set that has n elements, or equivalently. Outside of mathematics, the number also counts the number of different rhyme schemes for n-line poems. As well as appearing in counting problems, these numbers have a different interpretation, in particular, Bn is the nth moment of a Poisson distribution with mean 1. In general, Bn is the number of partitions of a set of size n, a partition of a set S is defined as a set of nonempty, pairwise disjoint subsets of S whose union is S. For example, B3 =5 because the 3-element set can be partitioned in 5 distinct ways, b0 is 1 because there is exactly one partition of the empty set. Every member of the empty set is a nonempty set, therefore, the empty set is the only partition of itself. As suggested by the set notation above, we consider neither the order of the partitions nor the order of elements within each partition and this means that the following partitionings are all considered identical. If, instead, different orderings of the sets are considered to be different partitions, If a number N is a squarefree positive integer, then Bn gives the number of different multiplicative partitions of N. These are factorizations of N into numbers greater than one, treating two factorizations as the same if they have the same factors in a different order. A rhyme scheme describes which lines rhyme with other. Thus, the 15 possible four-line rhyme schemes are AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, and ABCD. The Bell numbers come up in a card shuffling problem mentioned in the addendum to Gardner, of these, the number that return the deck to its original sorted order is exactly Bn. Thus, the probability that the deck is in its original order after shuffling it in this way is Bn/nn, probability that would describe a uniformly random permutation of the deck. Related to card shuffling are several problems of counting special kinds of permutations that are also answered by the Bell numbers. For instance, the nth Bell number equals number of permutations on n items in which no three values that are in sorted order have the last two of three consecutive. The permutations that avoid the generalized patterns 12-3, 32-1, 3-21, 1-32, 3-12, 21-3, the permutations in which every 321 pattern can be extended to a 3241 pattern are also counted by the Bell numbers

12.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing