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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.