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p^2 +qt =(p + t)^n, p^2 + qr = t^4, diophantine with primes

Source: 2021 JBMO TST Bosnia and Herzegovina P2

October 7, 2022
number theoryDiophantine equationdiophantine

Problem Statement

Let p,q,rp, q, r be prime numbers and t,nt, n be natural numbers such that p2+qt=(p+t)np^2 +qt =(p + t)^n and p2+qr=t4p^2 + qr = t^4 . a) Show that n<3n < 3. b) Determine all the numbers p,q,r,t,np, q, r, t, n that satisfy the given conditions.
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