In a paper I am writing, I have seven abstract statements and, corresponding to the first one, I have a theorem statement that says I will prove that there is a Pythagorean triplet for every pair of natural numbers $n,k$. I am using functions: $$A(n,k)=(2n-1)^2+2(2n-1)k$$ $$B(n,k)=2(2n-1)k+2k^2$...