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Diophantine Equation with prime numbers and bonus conditions

Source: 2023 Bulgaria JBMO TST Problem 3

May 7, 2023
number theoryprime numbersDiophantine equation

Problem Statement

Find all natural numbers aa, bb, cc and prime numbers pp and qq, such that:
■\blacksquare 4∤c4\nmid c ■\blacksquare p≢11(mod16)p\not\equiv 11\pmod{16} ■\blacksquare paqb−1=(p+4)cp^aq^b-1=(p+4)^c
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