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 By the early century, it had been superseded by number theory.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.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set
4.
3 (number)
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3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
5.
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
6.
7 (number)
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7 is the natural number following 6 and preceding 8. Seven, the prime number, is not only a Mersenne prime. It is also a Newman–Shanks–Williams prime, a Woodall prime, a prime, a lucky prime, a happy number, a safe prime. Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers, Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree. N =7 is the first natural number for which the statement does not hold, Two nilpotent endomorphisms from Cn with the same minimal polynomial. 7 is the only number D for which the equation 2n − D = x2 has more than two solutions for n and x natural, in particular, the equation 2n −7 = x2 is known as the Ramanujan–Nagell equation. 7 is the dimension, besides the familiar 3, in which a vector cross product can be defined. 7 is the lowest dimension of an exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere. 999,999 divided by 7 is exactly 142,857, for example, 1/7 =0.142857142857. and 2/7 =0.285714285714. In fact, if one sorts the digits in the number 142857 in ascending order,124578, the remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example,628 ÷7 =89 5/7, here 5 is the remainder, so in this case,628 ÷7 =89.714285. Another example,5238 ÷7 =748 2/7, hence the remainder is 2, in this case,5238 ÷7 =748.285714. A seven-sided shape is a heptagon, the regular n-gons for n ≤6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons are called heptagonal numbers, Seven is also a centered hexagonal number. Seven is the first integer reciprocal with infinitely repeating sexagesimal representation, There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers. There are seven types of catastrophes. When rolling two standard six-sided dice, seven has a 6 in 36 probability of being rolled, the greatest of any number, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved, in quaternary,7 is the smallest prime with a composite sum of digits
7.
19 (number)
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19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
8.
89 (number)
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89 is the natural number following 88 and preceding 90. 89 is, the 24th prime number, following 83 and preceding 97, the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms. An Eisenstein prime with no part and real part of the form 3n −1. A Fibonacci number and thus a Fibonacci prime as well, the first few digits of its reciprocal coincide with the Fibonacci sequence due to the identity 189 = ∑ n =1 ∞ F ×10 − =0.011235955 …. A Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers, M89 is the 10th Mersenne prime. Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse, among the known non-Lychrel numbers in the first 10000 integers, no other number requires that many or more iterations. The palindrome reached is also unusually large, eighty-nine is, The atomic number of actinium. Messier object M89, a magnitude 11.5 elliptical galaxy in the constellation Virgo, the New General Catalogue object NGC89, a magnitude 13.5 peculiar spiral galaxy in the constellation Phoenix and a member of Roberts Quartet. The Oklahoma Redhawks, an American minor league team, were formerly known as the Oklahoma 89ers. The number alludes to the Land Run of 1889, when the Unassigned Lands of Oklahoma were opened to white settlement, the teams home of Oklahoma City was founded during this event. In Rugby, an 89 or eight-nine move is a following a scrum, in which the number 8 catches the ball. The Elite 89 Award is presented by the U. S. NCAA to the participant in each of the NCAAs 89 championship finals with the highest grade point average. The jersey number 89 has been retired by three National Football League teams in honor of past playing greats, The Baltimore Colts, for Hall of Famer Gino Marchetti, the franchise continues to honor the number in its current identity as the Indianapolis Colts. The Boston Patriots, for Bob Dee, the franchise, now the New England Patriots, continues to honor the number. The Chicago Bears, for Mike Ditka, eighty-nine is also, The designation of Interstate 89, a freeway that runs from New Hampshire to Vermont The designation of U. S. The number of units of each colour in the board game Blokus The number of the French department Yonne Information Is Beautiful cites eighty-nine as one of the words censored on the Chinese internet
9.
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
10.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
11.
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
12.
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
13.
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
14.
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
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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
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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
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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