1.
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

2.
Square (algebra)
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In mathematics, a square is the result of multiplying a number by itself. The verb to square is used to denote this operation, squaring is the same as raising to the power 2, and is denoted by a superscript 2, for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the adjective which corresponds to squaring is quadratic. The square of an integer may also be called a number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, for instance, the square of the linear polynomial x +1 is the quadratic polynomial x2 + 2x +1. One of the important properties of squaring, for numbers as well as in other mathematical systems, is that. That is, the function satisfies the identity x2 =2. This can also be expressed by saying that the function is an even function. The squaring function preserves the order of numbers, larger numbers have larger squares. In other words, squaring is a function on the interval. Hence, zero is its global minimum, the only cases where the square x2 of a number is less than x occur when 0 < x <1, that is, when x belongs to an open interval. This implies that the square of an integer is never less than the original number, every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of one number, itself. For this reason, it is possible to define the square root function, no square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. There are several uses of the squaring function in geometry. The name of the squaring function shows its importance in the definition of the area, the area depends quadratically on the size, the area of a shape n times larger is n2 times greater. The squaring function is related to distance through the Pythagorean theorem and its generalization, Euclidean distance is not a smooth function, the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance, which has a paraboloid as its graph, is a smooth, the dot product of a Euclidean vector with itself is equal to the square of its length, v⋅v = v2

3.
5 (number)
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5 is a number, numeral, and glyph. It is the number following 4 and preceding 6. Five is the prime number. Because it can be written as 221 +1, five is classified as a Fermat prime, therefore a regular polygon with 5 sides is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also the number that is part of more than one pair of twin primes. Five is conjectured to be the only odd number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being 2 plus 3,5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation. Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers,5 is the length of the hypotenuse of the smallest integer-sided right triangle. In bases 10 and 20,5 is a 1-automorphic number,5 and 6 form a Ruth–Aaron pair under either definition. There are five solutions to Známs problem of length 6 and this is related to the fact that the symmetric group Sn is a solvable group for n ≤4 and not solvable for n ≥5. While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar, K5, Five is also the number of Platonic solids. A polygon with five sides is a pentagon, figurate numbers representing pentagons are called pentagonal numbers. Five is also a square pyramidal number, Five is the only prime number to end in the digit 5, because all other numbers written with a 5 in the ones-place under the decimal system are multiples of five. As a consequence of this,5 is in base 10 a 1-automorphic number, vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions, unlike expansions with all other prime denominators, because they are prime factors of ten, the base. When written in the system, all multiples of 5 will end in either 5 or 0

4.
17 (number)
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17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution

5.
37 (number)
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37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37

6.
53 (number)
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53 is the natural number following 52 and preceding 54. Fifty-three is the 16th prime number and it is also an Eisenstein prime, and a Sophie Germain prime. The sum of the first 53 primes is 5830, which is divisible by 53,53 written in hexadecimal is 35, that is, the same characters used in the decimal representation, but reversed. Four multiples of 53 share this property,371 =17316,5141 =141516,99481 =1849916, and 8520280 =82025816,53 cannot be expressed as the sum of any integer and its base-10 digits, making 53 a self number. 53 is the smallest prime number that does not divide the order of any sporadic group, the duration of Saros series 53 was 1514.5 years, and it contained 85 solar eclipses. The Saros number of the lunar eclipse series began on June 5,993 BC. The duration of Saros series 53 was 1280.1 years, fictional 53rd Precinct in the Bronx was found in the TV comedy Car 54, Where Are You. UDP and TCP port number for the Domain Name System protocol, 53-TET is a musical temperament that has a fifth that is closer to pure than our current system. 53 More Things To Do In Zero Gravity is a mentioned in The Hitchhikers Guide to the Galaxy. 53 a number used on the hand of the tulip in Infinity Train

7.
59 (number)
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59 is the natural number following 58 and preceding 60. Fifty-nine is the 17th smallest prime number, the next is sixty-one, with which it comprises a twin prime. 59 is an prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no part and real part of the form 3n −1. +1 is divisible by 59 but 59 is not one more than a multiple of 15,59 is a Pillai prime and it is also a highly cototient number. There are 59 stellations of the icosahedron,59 is one of the factors that divides the smallest composite Euclid number. In this case 59 divides the Euclid number 13# +1 =2 ×3 ×5 ×7 ×11 ×13 +1 =59 ×509, the duration of Saros series 59 was 1280.1 years, and it contained 72 solar eclipses. The Saros number of the lunar eclipse series began in March,729 BC

8.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0

9.
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

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.
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

12.
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

13.
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

14.
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

15.
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