Prove that $m^2+n^2=q^2\iff m=3,n=4,q=5$

Question

Hi

Can someone please give me an idea where to start with this ?

Thanks Steve

Answer 1

$$ m^2 + n^2 = q^2 \implies m^2 + (m+1)^2 = (m+2)^2 $$ Expanding, $$ 2m^2 + 2m + 1 = m^2 + 4m + 4 \implies m^2 - 2m - 3 = 0 $$ What does this polynomial tell you?

Answer 2

$$ m^2 + n^2 = q^2 \implies (n-1)^2 + n^2 = (n+1)^2 $$ which simplifies to $$ n^2=4n $$