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

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.