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Weird conditions on a number

Source: Czech and Slovak Olympiad 2015, National Round, Problem 1

April 1, 2015
number theorynational olympiadDiophantine equation

Problem Statement

Find all 4-digit numbers nn, such that n=pqrn=pqr, where p<q<rp<q<r are distinct primes, such that p+q=rāˆ’qp+q=r-q and p+q+r=s2p+q+r=s^2, where ss is a prime number.
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