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I have a related problem.

Consider the equation f(z,y) = [tex][(z^5-(z-y)^5)-y^5]/5[/tex]

which reduces to [tex]zy(z^3 -2z^2y +2zy^2 -y^3)[/tex]

If z is congruent to y, the congruent sum of the first and last terms within the ( ) is congruent to 0 mod 5, and likewise for the second and third terms, with the result that

f(z,y) is congruent to 0 mod 5 if z is congruent to y mod 5.

By brute force, I can prove that if z is not congruent to y mod 5 that

f(z,y) is not congruent to 0 mod 5. This involved substituting all the possible congruent combinations values of z and y, adding them up, and determining that they never equal 0.

My question is, can one derive a general mathmetical proof of this?

Better still, can one derive a general proof for this if we use p=any prime number, instead of 5?