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Power of k residues not in arithmetic progression

Source: China TST 4 2018 Day 2 Q4

July 17, 2018
modular arithmeticarithmetic sequencenumber theory

Problem Statement

Let pp be a prime and kk be a positive integer. Set SS contains all positive integers aa satisfying 1≤a≤p−11\le a \le p-1, and there exists positive integer xx such that xk≡a(modp)x^k\equiv a \pmod p. Suppose that 3≤∣S∣≤p−23\le |S| \le p-2. Prove that the elements of SS, when arranged in increasing order, does not form an arithmetic progression.
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