If f∈ℤ[x,y] generete only prime, then f is constant.

Suppose f is not constant, then, since ℤ[x,y]=(ℤ[y])[x] we can write:

f=a+f₁(y)x + … + fk(y)xᵏ

where fi ∈ ℤ[y] and fi≠0 for every i=1,…,k. I claim that there is y0∈ℤ such that f(x,y0) is not constant. Indeed if for every y0 in ℤ f(x,y0) is constant, then for every i=1,…,k fi(y) is zero in every point of ℤ, and so fi=0 for every i=1,…,k, contradicting the fact that f is not constant. So there is y0∈ℤ such that f(x,y0) is not constant. However g(x)=f(x,y0) is a polynomial with coefficient in ℤ that generete only prime numbers so f(x,y0) is constant, contradiction. □