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Equation has non-trivial solution iff $m=p$ (Balkan 1993)

Source: Balkan MO 1993, Problem 4

April 25, 2006
number theory proposednumber theory

Problem Statement

Let pp be a prime and m≥2m \geq 2 be an integer. Prove that the equation xp+yp2=(x+y2)m \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m has a positive integer solution (x,y)≠(1,1)(x, y) \neq (1, 1) if and only if m=pm = p.
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