All Questions

Are there any solutions in positive integers of $x^3 + 1 = (x - k) y^3$? The closest I can get is $19^3 + 1 = 20 \times 7^3$, but $20\gt 19$ so it just misses! For the related $x^3 - 1 = (x - k) y^3$,...

Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$. Q1 How small $u_k$ can be infinitely often as function $k$? This ...

In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$ see the reference Wikipedia Lehmer mean. The ...