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Source: Russian TST 2017, Day 7 P3 (Group NG)

April 1, 2023
number theoryresidue

Problem Statement

Let a1,…,ap−2a_1,\ldots , a_{p-2}{} be nonzero residues modulo an odd prime pp{}. For every d∣p−1d\mid p - 1 there are at least ⌊(p−2)/d⌋\lfloor(p - 2)/d\rfloor indices ii{} for which pp{} does not divide aid−1a_i^d-1. Prove that the product of some of a1,…,ap−2a_1,\ldots , a_{p-2} gives the remainder two modulo pp{}.
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