Let n be the smallest positive integer such that n(n+1)(n+2)(n+3) can be expressed as either a perfect square or a perfect cube (not necessarily both).
Find n, or prove that this is not possible.
The product can be rewritten as (n^2+3n+1)^2 - 1. This proves it is not a square unless n=0, -1, -2, or -3. None of those values are positive integers which the problem asks for.
The product cannot be a perfect cube either. If it was, then the values n*(n+1)*(n+2)*(n+3) and (n^2+3n+1)^2 would satisfy a special case of Catalan's Conjecture (case with p,q = 2,3).
It has been proven that the only nontrivial solution for the special case is 8=2^3 and 9=3^2. 8, 9 implies n^2+3n+1 = 3, which has no rational roots.
Edited on February 24, 2004, 3:41 pm
Solution (?)
