Advanced Search

S. S. Pillai proved that for a fixed positive integer $a$, the exponential Diophantine equation $x^y-y^x=a$, $min(x,y)>1$, has only finitely many solutions in integers $x$ and $y$. We prove that when $a$ is of the form $2z^2$, the above equation has no solution in integers $x$ and $y$ with $gcd(x,y)=1$.

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}\left(\sqrt{x^2-2y^n}\right)$ whose ideal class group has an element of order $n$. This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups.

We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a{13}^b{17}^c=2^m{y}^n$, where $a,b,c,m\ge 0,n\ge 3$ and $x,y\ge 1$ are unknown integers with $gcd(x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves.