Question 3
Positive integers $a_1,a_2,\ldots$ satisfy $a_1=2$ and
$$a_{n+1}=a_n^{n+1}-1\qquad(n\ge1).$$
Prove that for every odd prime $p$ and integer $k$, some term is divisible by $p^k$.
Proposed by John Berman. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30216450. One editorial point; manual grading.
$$a_{n+1}=a_n^{n+1}-1\qquad(n\ge1).$$
Prove that for every odd prime $p$ and integer $k$, some term is divisible by $p^k$.
Proposed by John Berman. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30216450. One editorial point; manual grading.