As a by-product of the negative solution of Hilbert’s 10th problem, various prime-generating polynomials were found. The best known upper bound for the number of variables in such a polynomial, to wit 10, was found by Yuri V. Matiyasevich in 1977. We show that this bound could be lowered to 8 if the converse of Wolstenholme’s theorem (1862) holds, as conjectured by James P. Jones. This potential improvement is achieved through a Diophantine representation of the set of all integers p >= 5 that satisfy the congruence C(2 p,p) ≡ 2 mod p^3. Our specification, in its turn, relies upon a terse polynomial representation of exponentiation due to Matiyasevich and Julia Robinson (1975), as further manipulated by Maxim Vsemirnov (1997). We briefly address the issue of also determining a lower bound for the number of variables in a prime-representing polynomial, and discuss the autonomous significance of our result about Wostenholme’s pseudoprimality, independently of Jones’s conjecture.

### A Diophantine representation of Wolstenholme's pseudoprimality

#### Abstract

As a by-product of the negative solution of Hilbert’s 10th problem, various prime-generating polynomials were found. The best known upper bound for the number of variables in such a polynomial, to wit 10, was found by Yuri V. Matiyasevich in 1977. We show that this bound could be lowered to 8 if the converse of Wolstenholme’s theorem (1862) holds, as conjectured by James P. Jones. This potential improvement is achieved through a Diophantine representation of the set of all integers p >= 5 that satisfy the congruence C(2 p,p) ≡ 2 mod p^3. Our specification, in its turn, relies upon a terse polynomial representation of exponentiation due to Matiyasevich and Julia Robinson (1975), as further manipulated by Maxim Vsemirnov (1997). We briefly address the issue of also determining a lower bound for the number of variables in a prime-representing polynomial, and discuss the autonomous significance of our result about Wostenholme’s pseudoprimality, independently of Jones’s conjecture.
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2015
http://ceur-ws.org/Vol-1459/
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11368/2846750`
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