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powerful numbers

Source: Canada 2005

June 26, 2009
number theory unsolvednumber theory

Problem Statement

Let's say that an ordered triple of positive integers (a,b,c)(a,b,c) is nn-powerful if a≤b≤c,gcd⁡(a,b,c)=1a\le b\le c,\gcd (a,b,c)=1 and an+bn+cna^n+b^n+c^n is divisible by a+b+ca+b+c. For example, (1,2,2)(1,2,2) is 55-powerful. a)a) Determine all ordered triples (if any) which are nn-powerful for all n≥1n\ge 1. b)b) Determine all ordered triples (if any) which are 20042004-powerful and 20052005-powerful, but not 20072007-powerful.
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