MathNet 2014 Question 85

Let $k$ be a fixed odd integer, $k \gt 3$. 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 $d_1 + d_2 = n + k$.