On the Diophantine complexity of the set of prime numbers

The ‘positive aspects of a negative solution’ (the recursive unsolvability of Hilbert’s 10th problem) include the discovery of Diophantine representations of the set P of primes. What is the rank of P, namely the smallest possible number, r, of unknowns in a polynomial representing P ? Siegel’s theorem on integral points on curves (1929) hands us a revealing characterisation of the Diophantine subsets of Z which can be represented in terms of a single unknown; thereby, since 19th century results about the density of P entail that P does not meet that characterisation, we get the lower rank bound r >=2. We also show that the Diophantine set consisting of those integers κ > 3 which meet the congruence (2 κ choose k) ≡ 2 mod κ^3 has rank not exceeding 7. As a consequence, the least known upper rank bound for P, namely r<=9 as found by Yu. V. Matiyasevich in 1977, can be lowered to r<=7 if the converse of Wolstenholme’s theorem (1862) holds, as conjectured by J. P. Jones.