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JBMO 2015 Problem 1

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Find all prime numbers a,b,ca, b, c and positive integers kk which satisfy the equation

a2+b2+16⋅c2=9⋅k2+1a^{2}+b^{2}+16 \cdot c^{2}=9 \cdot k^{2}+1

Dataset text NuminaMath olympiads. Quoted with source.

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(37,3,3,13),(17,3,3,7),(3,37,3,13),(3,17,3,7),(3,3,2,3)

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