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primes p,q in the form x^2 + 2005y^2, (x, y \in Z)

Source: KJMO 2005 p6

May 1, 2019
number theoryprimesset theory

Problem Statement

For two different prime numbers p,qp, q, defi ne Sp,q={p,q,pq}S_{p,q} = \{p,q,pq\}. If two elements in Sp,qS_{p,q} are numbers in the form of x2+2005y2,(x,y∈Z)x^2 + 2005y^2, (x, y \in Z), prove that all three elements in Sp,qS_{p,q} are in such form.
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