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Residues over a "quite" large prime.

Source: Vietnam TST 2021 P6

April 2, 2021
number theoryprime numbers

Problem Statement

Let n≥3n \geq 3 be a positive integers and pp be a prime number such that p>6n−1−2n+1p > 6^{n-1} - 2^n + 1. Let SS be the set of nn positive integers with different residues modulo pp. Show that there exists a positive integer cc such that there are exactly two ordered triples (x,y,z)∈S3(x,y,z) \in S^3 with distinct elements, such that x−y+z−cx-y+z-c is divisible by pp.
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