Formula Based On A System of Diophantine Equations
A system of 14 Diophantine equations in 26 variables can be used to obtain a Diophantine representation of the set of all primes. Jones et al. (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 numbers:
- α0 = = 0
- α1 = = 0
- α2 = = 0
- α3 = = 0
- α4 = = 0
- α5 = = 0
- α6 = = 0
- α7 = = 0
- α8 = = 0
- α9 = = 0
- α10 = = 0
- α11 = = 0
- α12 = = 0
- α13 = = 0
The 14 equations α0, …, α13 can be used to produce a prime-generating polynomial inequality in 26 variables:
i.e.:
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 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 1045). On the other hand, there also exists such a set of equations of degree only 4, but in 58 variables.(Jones 1982)
Read more about this topic: Formula For Primes
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