henderson 31.10.2015 15:45 Let $n$, $m$ and $k$ be positive integers satisfying $(n-1)n(n+1)=m^k.$ Prove that $k=1.$
Gryphos 31.10.2015 16:44 This is from Baltic Way 2012: http://artofproblemsolving.com/community/c6h508361p2855974
MilosMilicev 17.10.2016 19:37 Since n and n^2-1 are relatively prime, we obtain that n=a^k, n^2-1=b^k, a,b are positive integers. It means that (a^2)^k-b^k=1 ==> a^2>b. Let a^2=b+s, s is natural. (b+s)^k-b^k>(b+s)-b=s>=1 (for k>1). We get that k=1.