On arithmetic progressions of positive integers avoiding $p+F_m$ and $q+L_n$

TL;DR

This paper proves the existence of an arithmetic progression of positive integers that cannot be expressed as either a prime plus a Fibonacci number or a prime plus a Lucas number, revealing new structural properties of these number sets.

Contribution

It establishes the existence of such an arithmetic progression, a novel result linking primes, Fibonacci, and Lucas numbers.

Findings

Existence of an arithmetic progression avoiding p+F_m and q+L_n

New insights into the distribution of primes, Fibonacci, and Lucas numbers

Advances understanding of additive properties of special number sets

Abstract

In this paper, it is proved that there is an arithmetic progression of positive integers such that each of which is expressible neither as $p+F_m$ nor as $q+L_n$, where $ p,q $ are primes, $ F_m $ denotes the $ m $-th Fibonacci number and $ L_n $ denotes the $ n $-th Lucas number.