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on $S = \{m^2 + dn^2 \}

Source: 11-th Hungary-Israel Binational Mathematical Competition 2000

April 22, 2007
number theory proposednumber theory

Problem Statement

For a given integer dd, let us define S={m2+dn2∣m,n∈Z}S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}. Suppose that p,qp, q are two elements of SS , where pp is prime and p∣qp | q. Prove that r=q/pr = q/p also belongs to SS .
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