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Looking for n (Posted on 2004-02-24) Difficulty: 3 of 5
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.

See The Solution Submitted by Aaron    
Rating: 4.2857 (7 votes)

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Solution (?) | Comment 2 of 12 |

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
  Posted by Brian Smith on 2004-02-24 15:36:24

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