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Seemingly ugly NT with 4 conditions

Source: Serbia JBMO TST 2023 P4

May 28, 2023
number theory

Problem Statement

Find all triples (k,m,n)(k, m, n) of positive integers such that mm is a prime and:
(1) knkn is a perfect square;
(2) k(k−1)2+n\frac{k(k-1)}{2}+n is a fourth power of a prime;
(3) k−m2=pk-m^2=p where pp is a prime;
(4) n+2m2=p4\frac{n+2}{m^2}=p^4.
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