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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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Not a cube | Comment 9 of 13 |
Euler showed that the only solution of y2 - x3 = ±1 is (x,y) = (2,3), a special case of Catalan's Conjecture, which has recently been proved.

Using Euler's result, n(n+1)(n+2)(n+3) = (n2+3n+1)2 - 1 cannot be a perfect cube.  However, there must be a more elementary solution!

Incidentally, Erdos and Selfridge proved that the product of any number of consecutive positive integers is never a perfect power.

Edited on February 29, 2004, 11:03 am
  Posted by Nick Hobson on 2004-02-29 11:00:33

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