Editing Formula for primes (section)

==Formula based on a system of Diophantine equations==
Because the set of primes is a [[computably enumerable set]], by [[Matiyasevich's theorem]], it can be obtained from a system of [[Diophantine equation]]s. {{Harvtxt|Jones|Sato|Wada|Wiens|1976}} found an explicit set of 14 Diophantine equations in 26 variables, such that a given number ''k''&thinsp;+&thinsp;2 is prime [[if and only if]] that system has a solution in nonnegative integers:<ref>{{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 | url-status=dead | archive-url=https://web.archive.org/web/20120224013618/http://mathdl.maa.org/mathDL/?pa=content&sa=viewDocument&nodeId=2967&pf=1 | archive-date=2012-02-24 }}.</ref>

:  <math>\alpha_0= wz + h + j - q  = 0</math>

: <math>\alpha_1 = (gk + 2g + k + 1)(h + j) + h - z  = 0</math>

: <math>\alpha_2= 16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2  = 0</math>

: <math>\alpha_3= 2n + p + q + z - e  = 0</math>

: <math>\alpha_4= e^3(e + 2)(a + 1)^2 + 1 - o^2 = 0</math>

: <math>\alpha_5=(a^2 - 1)y^2 + 1 - x^2 = 0</math>

: <math>\alpha_6= 16r^2y^4(a^2 - 1) + 1 - u^2 = 0</math>

: <math>\alpha_7= n + \ell + v - y = 0</math>

: <math>\alpha_8= (a^2 - 1)\ell^2 + 1 - m^2 = 0</math>

: <math>\alpha_9= ai + k + 1 - \ell - i = 0</math>

: <math>\alpha_{10}= ((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2 = 0</math>

: <math>\alpha_{11}= p + \ell(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m= 0 </math>

: <math>\alpha_{12}= q + y(a - p - 1) + s(2ap + 2a - p^2 - 2p - 2) - x = 0</math>

:<math>\alpha_{13}= z + p\ell(a - p) + t(2ap - p^2 - 1) - pm = 0</math>

The 14 equations <math>\alpha_0, \dots, \alpha_{13}</math> can be used to produce a prime-generating polynomial inequality in 26 variables:

:<math> (k+2)(1-\alpha_0^2-\alpha_1^2-\cdots-\alpha_{13}^2) > 0.</math>

That is,

: <math>
\begin{align}
& (k+2) (1 - {} \\[6pt]
& [wz + h + j - q]^2 - {} \\[6pt]
& [(gk + 2g + k + 1)(h + j) + h - z]^2 - {} \\[6pt]
& [16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2 - {} \\[6pt]
& [2n + p + q + z - e]^2 - {} \\[6pt]
& [e^3(e + 2)(a + 1)^2 + 1 - o^2]^2 - {} \\[6pt]
& [(a^2 - 1)y^2 + 1 - x^2]^2 - {} \\[6pt]
& [16r^2y^4(a^2 - 1) + 1 - u^2]^2 - {} \\[6pt]
& [n + \ell + v - y]^2 - {} \\[6pt]
& [(a^2 - 1)\ell^2 + 1 - m^2]^2 - {} \\[6pt]
& [ai + k + 1 - \ell - i]^2 - {} \\[6pt]
& [((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^2 - {} \\[6pt]
& [p + \ell(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m]^2 - {} \\[6pt]
& [q + y(a - p - 1) + s(2ap + 2a - p^2 - 2p - 2) - x]^2 - {} \\[6pt]
& [z + p\ell(a - p) + t(2ap - p^2 - 1) - pm]^2) \\[6pt]
& > 0
\end{align}
</math>

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.

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.<ref>{{citation | last = Matiyasevich | first = Yuri V. | author-link = Yuri Matiyasevich | year=1999 | chapter = Formulas for Prime Numbers | chapter-url=https://books.google.com/books?id=oLKlk5o6WroC&pg=PA13 | editor1-first=Serge | editor1-last = Tabachnikov | editor-link1=Sergei Tabachnikov| title = Kvant Selecta: Algebra and Analysis | volume = II | publisher = [[American Mathematical Society]] | isbn = 978-0-8218-1915-9 | pages=13–24}}.</ref> Hence, there is a prime-generating polynomial inequality 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.<ref>{{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| jstor=2273588 | s2cid=11148823 }}.</ref>