Problem

Source: 2014 China TST 3 Day 2 Q4

Tags: number theory proposed, number theory, Vieta Jumping



Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.