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Special representation of primes and their squares

Source: Bulgaria MO Regional round 2024, 10.3

February 13, 2024
number theory

Problem Statement

Find all positive integers 1≤k≤61 \leq k \leq 6 such that for any prime pp, satisfying p2=a2+kb2p^2=a^2+kb^2 for some positive integers a,ba, b, there exist positive integers x,yx, y, satisfying p=x2+ky2p=x^2+ky^2.
[hide=Remark on 10.4] It also appears as ARO 2010 10.4 with the grid changed to 10×1010 \times 10 and 1717 changed to 55, so it will not be posted.
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