Image of an integer polynomial in the integers mod 7

Question

Given p(x,y) = x² - 7y² - 24 ∈ ℤ[x,y], does the image in ℤ₇[x,y] become x² - 3 or x² + 4? Or could I use either to determine whether or not p(x,y) has any solutions in ℤ[x,y]?

Answer 1

You could use either $x^2-3$ or $x^2+4$, or in fact $x^2+(4+n)\times 7$ for any $ n \in \mathbb N;$

adding any multiple of $7$ doesn't change things modulo $7$.