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Baltic Way competition problems, 2016

Number theory Difficulty 4.9 AIME Prove it Baltic Way

Determine all positive integers $a$ and all primes $p$ fulfilling the equation
$(a - p)^3 = a + p.$

Solution

Writing $n = a - p$ transforms the equation into
$n^3 = a + p = n + 2p,$
so that
$2p = n^3 - n = n(n + 1)(n - 1),$
which is divisible by $3$ . Therefore $p = 3$ and $n = 2$ , whence $a = 5$ .

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