Problem

Source: Iranian 3rd round Number Theory exam P4

Tags: number theory proposed, number theory



$2 \leq d$ is a natural number. $B_{a,b}$={$a,a+b,a+2b,...,a+db$} $A_{c,q}$={$cq^n \vert n \in\mathbb{N}$} Prove that there are finite prime numbers like $p$ such exists $a,b,c,q$ from natural numbers : $i$ ) $ p \nmid abcq $ $ ii$ ) $A_{c,q} \equiv B_{a,b} (mod p ) $ (15 points )