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.

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