Problem A. 916. (October 2025)

A. 916. Let \(\displaystyle a \geq 3\) be an integer, and define \(\displaystyle f(n) = a^n - 1\) for every positive integer \(\displaystyle n\). Denote by \(\displaystyle f^{(k)}\) the \(\displaystyle k\)-iterate of \(\displaystyle f\), that is, \(\displaystyle f^{(1)}(n) = f(n)\) and \(\displaystyle f^{(k+1)}(n)=f(f^{(k)}(n))\) for \(\displaystyle k \geq 1\).

a) Prove that for any positive integer \(\displaystyle K\) there exists a positive integer \(\displaystyle M\) such that for every integer \(\displaystyle 1 \leq k \leq K\), the number \(\displaystyle f^{(k)}(M)\) is divisible by \(\displaystyle M\) if and only if \(\displaystyle k\) is divisible by \(\displaystyle 2025\).

b) Does there exist a positive integer \(\displaystyle N\) such that for every positive integer \(\displaystyle k\), the number \(\displaystyle f^{(k)}(N)\) is divisible by \(\displaystyle N\) if and only if \(\displaystyle k\) is divisible by \(\displaystyle 2025\)?

Proposed by Boldizsár Varga, Budapest

(7 pont)

Deadline expired on November 10, 2025.

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Problems in Mathematics of KöMaL, October 2025