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| In [[number theory]], a '''formula for primes''' is a formula generating the [[prime number]]s, exactly and without exception. No such formula which is [[algorithmic_efficiency|efficiently computable]] is presently known. A number of constraints are known: what such a "formula" can and cannot be.
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| ==Prime formulas and polynomial functions==
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| It is known that no non-[[constant term|constant]] [[polynomial]] function ''P''(''n'') with integer coefficients exists that evaluates to a [[prime number]] for all integers ''n''. The proof is as follows: Suppose such a polynomial existed. Then ''P''(1) would evaluate to a prime ''p'', so <math>P(1) \equiv 0 \pmod p</math>. But for any ''k'', <math>P(1+kp) \equiv 0 \pmod p</math> also, so <math>P(1+kp)</math> cannot also be prime (as it would be divisible by ''p'') unless it were ''p'' itself, but the only way <math>P(1+kp) = P(1)</math> for all ''k'' is if the polynomial function is constant.
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| The same reasoning shows an even stronger result: no non-constant polynomial function ''P''(''n'') exists that evaluates to a prime number for [[almost all]] integers ''n''.
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| [[Leonhard Euler|Euler]] first noticed (in 1772) that the [[quadratic polynomial]]
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| :''P''(''n'') = ''n''<sup>2</sup> − ''n'' + 41
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| is prime for all [[positive integer]]s less than 41. The primes for ''n'' = 1, 2, 3... are 41, 43, 47, 53, 61, 71... The differences between the terms are 2, 4, 6, 8, 10... For ''n'' = 41, it produces a square number, 1681, which is equal to 41×41, the smallest [[composite number]] for this formula. If 41 divides ''n'' it divides ''P(n)'' too. The phenomenon is related to the [[Ulam spiral]], which is also implicitly quadratic, and the [[class number (number theory)|class number]]; | |
| this polynomial is related to the [[Heegner number]] <math>163=4\cdot 41-1</math>, and there are analogous polynomials for <math>p=2, 3, 5, 11, \text{ and } 17</math>, corresponding to other Heegner numbers. | |
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| It is known, based on [[Dirichlet's theorem on arithmetic progressions]], that linear polynomial functions <math>L(n) = an + b</math> produce infinitely many primes as long as ''a'' and ''b'' are [[relatively prime]] (though no such function will assume prime values for all values of ''n''). Moreover, the [[Green–Tao theorem]] says that for any ''k'' there exists a pair of ''a'' and ''b'' with the property that <math>L(n) = an+b</math> is prime for any ''n'' from 0 to ''k'' − 1. However, the best known result of such type is for ''k'' = 26:
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| :43142746595714191 + 5283234035979900''n'' is prime for all ''n'' from 0 to 25 {{harv|Andersen|2010}}.
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| It is not even known whether there exists a univariate polynomial of degree at least 2 that assumes an infinite number of values that are prime; see [[Bunyakovsky conjecture]].
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| ==Formula based on a system of Diophantine equations==
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| A system of 14 [[Diophantine equation]]s in 26 variables can be used to obtain a Diophantine representation of the set of all primes. {{Harvtxt|Jones|Sato|Wada|Wiens|1976}} proved that a given number ''k'' + 2 is prime [[if and only if]] the following system of 14 Diophantine equations has a solution in the [[natural number]]s:
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| : α<sub>0</sub> = <math> wz + h + j - q </math> = 0
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| : α<sub>1</sub> = <math> (gk + 2g + k + 1)(h + j) + h - z </math> = 0
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| : α<sub>2</sub> = <math> 16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2 </math> = 0
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| : α<sub>3</sub> = <math> 2n + p + q + z - e </math> = 0
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| : α<sub>4</sub> = <math> e^3(e + 2)(a + 1)^2 + 1 - o^2 </math> = 0
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| : α<sub>5</sub> = <math> (a^2 - 1)y^2 + 1 - x^2 </math> = 0
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| : α<sub>6</sub> = <math> 16r^2y^4(a^2 - 1) + 1 - u^2 </math> = 0
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| : α<sub>7</sub> = <math> n + l + v - y </math> = 0
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| : α<sub>8</sub> = <math> (a^2 - 1)l^2 + 1 - m^2 </math> = 0
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| : α<sub>9</sub> = <math> ai + k + 1 - l - i </math> = 0
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| : α<sub>10</sub> = <math> ((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2 </math> = 0
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| : α<sub>11</sub> = <math> p + l(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m </math> = 0
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| : α<sub>12</sub> = <math> q + y(a - p - 1) + s(2ap + 2a - p^2 - 2p - 2) - x </math> = 0
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| : α<sub>13</sub> = <math> z + pl(a - p) + t(2ap - p^2 - 1) - pm </math> = 0
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| The 14 equations α<sub>0</sub>, …, α<sub>13</sub> can be used to produce a prime-generating polynomial inequality in 26 variables:
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| :<math> (k+2)(1-\alpha_0^2-\alpha_1^2-\cdots-\alpha_{13}^2) > 0 </math>
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| i.e.:
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| :<math> (k+2) (1 - </math>
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| :<math> [wz + h + j - q]^2 - </math>
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| :<math> [(gk + 2g + k + 1)(h + j) + h - z]^2 - </math> | |
| :<math> [16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2 - </math>
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| :<math> [2n + p + q + z - e]^2 - </math>
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| :<math> [e^3(e + 2)(a + 1)^2 + 1 - o^2]^2 - </math>
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| :<math> [(a^2 - 1)y^2 + 1 - x^2]^2 - </math>
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| :<math> [16r^2y^4(a^2 - 1) + 1 - u^2]^2 - </math>
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| :<math> [n + l + v - y]^2 - </math> | |
| :<math> [(a^2 - 1)l^2 + 1 - m^2]^2 - </math>
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| :<math> [ai + k + 1 - l - i]^2 - </math>
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| :<math> [((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^2 - </math>
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| :<math> [p + l(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m]^2 - </math>
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| :<math> [q + y(a - p - 1) + s(2ap + 2a - p^2 - 2p - 2) - x]^2 - </math>
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| :<math> [z + pl(a - p) + t(2ap - p^2 - 1) - pm]^2) </math>
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| :<math> > 0 </math>
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| is a polynomial inequality in 26 variables, and the set of prime numbers is identical to the set of positive values taken on by the left-hand side as the variables ''a'', ''b'', …, ''z'' range over the nonnegative integers.
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| A general theorem of [[Yuri Matiyasevich|Matiyasevich]] says that if a set is defined by a system of Diophantine equations, it can also be defined by a system of Diophantine equations in only 9 variables. Hence, there is a prime-generating polynomial as above with only 10 variables. However, its degree is large (in the order of 10<sup>45</sup>). On the other hand, there also exists such a set of equations of degree only 4, but in 58 variables.{{harv|Jones|1982}}
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| ==Mills' formula==
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| The first such formula known was established in 1947 by [[W. H. Mills]], who proved that there exists a [[real number]] ''A'' such that
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| :<math>\lfloor A^{3^{n}}\;\rfloor</math>
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| is a prime number for all positive integers ''n''. If the [[Riemann hypothesis]] is true, then the smallest such ''A'' has a value of around 1.3063... and is known as [[Mills' constant]]. This formula has no practical value, because very little is known about the constant (not even whether it is [[rational number|rational]]), and there is no known way of calculating the constant without finding primes in the first place.
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| ==Recurrence relation==
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| Another prime generator is defined by the [[recurrence relation]]
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| :<math> a_n = a_{n-1} + \operatorname{gcd}(n,a_{n-1}), \quad a_1 = 7, </math>
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| where gcd(''x'', ''y'') denotes the [[greatest common divisor]] of ''x'' and ''y''. The sequence of differences ''a''<sub>''n'' + 1</sub> − ''a<sub>n</sub>'' starts with 1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1 {{OEIS|id=A132199}}. {{harvtxt|Rowland|2008}} proved that this sequence contains only ones and prime numbers.
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| ==See also==
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| * [[Generating primes]]
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| * [[Prime number theorem]], which gives an estimate for the number of primes in a range but does not identify specific primes.
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| ==References==
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| <div class="references-small">
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| * {{citation | first1=Jens Kruse | last1=Andersen | year=2010 | title=Primes in Arithmetic Progression Records | url=http://users.cybercity.dk/~dsl522332/math/aprecords.htm | accessdate=2010-04-13 }}.
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| * {{citation | first1=Adrian | last1=Bowyer | year=n.d. | pages=11761 | title=Formulae for Primes | journal=Eprint arXiv:math/0611761 | url=http://people.bath.ac.uk/ensab/Primes/ | accessdate=2008-07-09 | bibcode=2006math.....11761F |arxiv = math/0611761 }}.
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| * {{citation | first1=James P. | last1=Jones | first2=Daihachiro | last2=Sato | first3=Hideo | last3=Wada | first4=Douglas | last4=Wiens | author4-link=Douglas Wiens | url = http://mathdl.maa.org/mathDL/?pa=content&sa=viewDocument&nodeId=2967&pf=1 | year=1976 | title=Diophantine representation of the set of prime numbers | journal=[[American Mathematical Monthly]] | volume=83 | pages=449–464 | doi=10.2307/2318339 | jstor=2318339 | issue=6 | publisher=Mathematical Association of America}}.
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| * {{citation | doi=10.2307/2273588 | first=James P. | last=Jones | year=1982 | title=Universal diophantine equation | journal=Journal of Symbolic Logic | volume=47 | issue=3 | pages=549–571}}..
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| * {{Citation | doi=10.2307/2690354 | last1=Regimbal | first1=Stephen | title=An explicit Formula for the k-th prime number | jstor=2690354 | year=1975 | journal=Mathematics Magazine | volume=48 | pages=230–232 | issue=4 | publisher=Mathematical Association of America}}.
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| * {{Citation | last1=Rowland | first1=Eric S. | title=A Natural Prime-Generating Recurrence | url=http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Rowland/rowland21.html | year=2008 | journal=Journal of Integer Sequences | volume=11 | pages=08.2.8 | bibcode=2008JIntS..11...28R|arxiv = 0710.3217 }}.
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| * {{citation | last = Matiyasevich | first = Yuri V. | authorlink = Yuri Matiyasevich | title = Formulas for Prime Numbers | journal = Eprint arXiv:math/0611761 | pages = 11761 | year = 2006 | bibcode = 2006math.....11761F|arxiv = math/0611761 }} ({{citation | first = Serge | last = Tabachnikov | title = Kvant selecta: algebra and analysis | volume = 1 | publisher = AMS Bookstore | isbn = 978-0-8218-1915-9}}, [http://books.google.com/books?id=oLKlk5o6WroC&pg=PA13 pp 13.]).
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| </div>
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| {{reflist}}
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| ==External links==
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| * {{MathWorld|urlname=PrimeFormulas|title=Prime Formulas}}
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| * {{MathWorld|urlname=Prime-GeneratingPolynomial|title=Prime-Generating Polynomial}}
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| * {{MathWorld|urlname=MillsConstant|title=Mill's Constant}}
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| *A Venugopalan. ''Formula for primes, twinprimes, number of primes and number of twinprimes''. Proceedings of the Indian Academy of Sciences—Mathematical Sciences, Vol. 92, No 1, September 1983, pp. 49–52. Page [http://www.ias.ac.in/jarch/mathsci/92/00000050.pdf 49], [http://www.ias.ac.in/jarch/mathsci/92/00000051.pdf 50], [http://www.ias.ac.in/jarch/mathsci/92/00000052.pdf 51], [http://www.ias.ac.in/jarch/mathsci/92/00000053.pdf 52], [http://www.ias.ac.in/jarch/mathsci/93/00000068.pdf errata].
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| {{Prime number classes}}
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| [[Category:Prime numbers]]
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