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Polish MO finals, problem 6

Source:

April 10, 2008
number theory proposednumber theory

Problem Statement

Let S S be a set of all positive integers which can be represented as a^2 \plus{} 5b^2 for some integers a,b a,b such that a⊥b a\bot b. Let p p be a prime number such that p \equal{} 4n \plus{} 3 for some integer n n. Show that if for some positive integer k k the number kp kp is in S S, then 2p 2p is in S S as well. Here, the notation a⊥b a\bot b means that the integers a a and b b are coprime.
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