All Questions

I. The Diophantine equation, $$x^3+y^3+z^3 = 3w^3\tag1$$ with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...

Is the following conjecture true? Conjecture. If $r > s \ge 1$ are relatively prime integers such that \begin{equation} (r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1} \end{equation} then $r-s = 1$ ...

Let $p,a,b,x,y$ be positive integers where $p$ is an odd prime; $x$ and $y$ are odd; $p,x$ and $y$ are all coprime. I'm interested in finding examples of such numbers that satisfy this equation: \...

In page 2 of Mordell "Diophantion equations" it is given a necessary condition for the solvability of a Diophantine equation: Integer solutions of the inhomogeneous equation $f(x) = 0$ can exist only ...

I propose the following problem (Maybe it has a trivial solution): Let $n$ be a positive integer such that $$n\equiv1 \pmod 4.$$ Then the problem is to find a rational $x$ as a function of $n$ such ...