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2021 Turkey Team Selection Test, 1

Let $n$ be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.

2018 Middle European Mathematical Olympiad, 7

Tags: number theory , legendre symbol

Let $a_1,a_2,a_3,\cdots$ be the sequence of positive integers such that $$a_1=1 , a_{k+1}=a^3_k+1, $$ for all positive integers $k.$ Prove that for every prime number $p$ of the form $3l +2,$ where $l$ is a non-negative integer ,there exists a positive integer $n$ such that $a_n$ is divisible by $p.$

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