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All The Possible Remainders

Source: KöMaL A. 792

March 24, 2022
komalnumber theory

Problem Statement

Let p≥3p\geq 3 be a prime number and 0≤r≤p−3.0\leq r\leq p-3. Let x1,x2,…,xp−1+rx_1,x_2,\ldots,x_{p-1+r} be integers satisfying ∑i=1p−1+rxik≡r mod p\sum_{i=1}^{p-1+r}x_i^k\equiv r \bmod{p}for all 1≤k≤p−2.1\leq k\leq p-2. What are the possible remainders of numbers x2,x2,…,xp−1+rx_2,x_2,\ldots,x_{p-1+r} modulo p?p?
Proposed by Dávid Matolcsi, Budapest
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