Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?

Question

In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the positive values are precisely the primes) must have at least three variables has been proven. Alon suggested that perhaps the number was a typo, that all that is known is that (trivially) no univariate polynomial is prime-representing.

As of Jones 1982 [1, p. 550] the question of the existence of a universal Diophantine equation in two variables was open, so certainly it was not known that the number of variables for the special case of the primes was more than 2 at that time.

[1] James P. Jones, "Universal Diophantine equation", The Journal of Symbolic Logic 47:3 (1982), pp. 549-571.

Answer 1

Davis [1] writes that the two-variable case of universal Diophantine equations is still open as of 2006. (Ribenboim's book was published in 1996.) So the question of a prime-representing polynomial in two variables was (and, presumably, is) still open.

[1] Martin Davis, [FOM] Decidability of Diophantine equations, post to the FOM mailing list, December 14 2006.

Answer 2

Greckuk, Grechuk, & Wilcox [1] give an algorithm for solving all 2-variable 3-monomial Diophantine equations. Two-variable Diophantine equations with 4 or more monomials are still open as of 2023.

[1] Bogdan Grechuk, Tetiana Grechuk, Ashleigh Wilcox, Diophantine equations with three monomials, Journal of Number Theory Volume 253, December 2023, Pages 69-108.