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Pythagorean-Euler Theorem?

Source: 2022 IMOC N5

September 5, 2022
IMOCnumber theory

Problem Statement

Find all solution (p,r)(p,r) of the "Pythagorean-Euler Theorem" pp+(p+1)p+⋯+(p+r)p=(p+r+1)pp^p+(p+1)^p+\cdots+(p+r)^p=(p+r+1)^pWhere pp is a prime and rr is a positive integer.
Proposed by Li4 and Untro368
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