# Image of an integer polynomial in the integers mod 7

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]?

• Either will do, they are equivalent $\pmod 7$.
– lulu
Dec 10 '18 at 18:51
• Indeed, @lulu, I would have said that as polynomials over the field with seven elements, they are equal. Dec 11 '18 at 4:11

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 $$7$$.