On the sum of a prime and two Fibonacci numbers

TL;DR

This paper investigates the representation of positive integers as the sum of a prime and two Fibonacci numbers with squared indices, proving the existence of infinite solutions and analyzing their distribution.

Contribution

It establishes that the set of integers with no such representation contains an infinite arithmetic progression, and that the sets with one or multiple solutions have positive densities.

Findings

The set of integers with no solutions contains an infinite arithmetic progression.

The sets with exactly one or multiple solutions have positive asymptotic densities.

The paper proves the existence of infinitely many solutions for the representation.

Abstract

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.