prove that 5n+3, where n is a prime, can never be a perfect square
So you want to prove that there is no x such that , that is
If x=0(mod5), then x≤=0(mod5)
If x=1(mod5), then x≤=1(mod5)
If x=2(mod5), then x≤=4(mod5)
If x=3(mod5), then x≤=4(mod5)
If x=4(mod5), then x≤=1(mod5)
thus it x doesn't exist.
Hmm if you want to apply FLT, it's actually possible.
Assume there is x such that
Square it :
But we know, by FLT, that for any x coprime with 5,
And if it is not coprime with 5, it means that it's a multiple of 5. In which case,
So we have a contradiction and there is no x such that
Is it more similar to what you're doing in class ?