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

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.