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Sum of powers

Source: Turkish TST 2012 Problem 7

March 26, 2012
limitcalculusintegrationfunctionalgebra proposedalgebra

Problem Statement

Let Sr(n)=1r+2r+⋯+nrS_r(n)=1^r+2^r+\cdots+n^r where nn is a positive integer and rr is a rational number. If Sa(n)=(Sb(n))cS_a(n)=(S_b(n))^c for all positive integers nn where a,ba, b are positive rationals and cc is positive integer then we call (a,b,c)(a,b,c) as nice triple. Find all nice triples.
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