Proof of no prime-representing polynomial in 2 variables

Question

In "The New Book of Prime Number Records", Ribenboim reviews the known results on the degree and number of variables of prime-representing polynomials (those are polynomials such that the set of positive values they obtain for nonnegative integral values of the variables coincides with the set of primes). For example, it is known that there is such a polynomial with 42 variables and degree 5, as well as one with 10 variables and astronomical degree.

Ribenboim mentions that it's an open problem to determine the least number of variables possible for such a polynomial, and remarks "it cannot be 2". It's a fairly simple exercise to show that it cannot be 1, but why can't it be 2?

EDIT: here's the relevant excerpt from Ribenboim's book. Given that nobody seems to be familiar with such a proof, I'm inclined to assume that this is a typo and he just meant "it cannot be 1".

Answer 1

I asked the same question at MathOverflow (linking here) where I noted that, at least as of 1982, the problem was still open because even universal Diophantine equations were not known to be impossible with two variables.

On further searching I found a FOM posting (see link above) which shows that the universal Diophantine equation problem is still open, so it looks like Ribenboim's book is in error (probably a typo, as Alon suggests).

Answer 2

Given all the evidence so far, I'm inclined to declare Ribenboim's parenthetic remark a typo. He probably meant either "it cannot be 1" or "it must be at least 2". It would be confusing and unusual for him to mention this off-hand with no reference as if it's a simple observation. That it certainly isn't.