All Questions

While I was working on a paper on graph theory, I encountered a problem which I think is a number-theory-problem. I don't know if there are any tools to answer the question. Find all natural numbers $...

I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation $$ x^2+y^...

I am looking for infinite set of Diophantine solutions. Suppose we require $$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$ $$a,b,c,d\in\mathbb Z$$ then can we still find ...