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All values of a quadratic are semiprime

Source: 2023 Grosman MO, P4

September 8, 2023
quadraticsnumber theoryprime numbers

Problem Statement

Let qq be an odd prime number. Prove that it is impossible for all (q−1)(q-1) numbers 12+1+q,22+2+q,…,(q−1)2+(q−1)+q1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q to be products of two primes (not necessarily distinct).
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