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Number theory Difficulty 7.7 National Olympiad, round 2 Prove it Baltic Way

Let $p \neq 3$ be a prime number. Show that there is a non-constant arithmetic sequence of positive integers $x_1, x_2, \dots, x_p$ such that the product of the terms of the sequence is a cube.

Solution

Let $a_1, a_2, \dots, a_p$ be any arithmetic sequence of positive integers and let $P$ be the product of the terms of this sequence. For any $n$ , the sequence $P^n a_1, P^n a_2, \dots, P^n a_p$ is also arithmetic, and the product of terms is $P^{np+1}$ . Now either $p \equiv 1 \pmod 3$ or $p \equiv -1 \pmod 3$ . In the former case, $2p+1 = 3q$ for some $q$ and in the latter case, $1p+1 = 3q$ for some $q$ . So we can choose either $x_i = P^2 a_i$ or $x_i = P a_i$ to obtain the sequence we are looking for.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.