Let $\{ f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{ k_1^{ 2}} +f_{ k_{ 2}^{ 2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper, it is proved that $\{ n : r(n)=0\} $ contains an infinite arithmetic progression, and both sets $\{ n : r(n)=1\} $ and $\{ n : r(n)\ge 2\}$ have positive asymptotic densities.
Categories: math.NT