Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b\equiv 0\left(mod{2}^r\right)$ and $b\equiv \pm 2^r\left(moda\right)$ for some non-negative integer r, then the Diophantine equation $a^x+b^y=c^z$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).

We determine the distribution over square-free integers $n$ of the pair $({dim}_{{𝔽}_2}{\mathrm{Sel}}^{\Phi }(E_n/\mathbb{Q}),{dim}_{{𝔽}_2}{\mathrm{Sel}}^{\widehat{\Phi }}(E_n^{\text{'}}/\mathbb{Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n^{\text{'}}:y^2=x^3+4n^{2}x$ is the image of isogeny $\Phi :E_n\to E_n^{\text{'}}$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat{\Phi }$ is the isogeny dual to $\Phi$.

It is known that two consecutive coefficients of a ternary cyclotomic polynomial ${\Phi }_{pqr}\left(x\right)={\sum }_k{a}_{pqr}\left(k\right)x^k$ differ by at most one. We characterize all k such that $|a_{pqr}\left(k\right)-a_{pqr}(k-1)|=1$. We use this to prove that the number of nonzero coefficients of the nth ternary cyclotomic polynomial is greater than $n^{1/3}$.