Prove that the product of four consecutive positive integers cannot be a perfect square.
If n is the smallest of the four integers, then the product is
(n)(n+1)(n+2)(n+3) = (n2 + 3n)(n2 + 3n + 2)
= (n2 + 3n + 1)2 – 1
This can’t be a perfect square, because two positive squares cannot differ by 1.
From Angela Dunn’s Mathematical Bafflers, via David Wells’ Penguin Book of Curious and Interesting Puzzles, 1992.
March 13, 2013 | Puzzles, Science & Math
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