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Multiplying gets a square? (Posted on 2004-06-26) Difficulty: 3 of 5
Is it possible to get a perfect square if you multiply three consecutive natural numbers?

See The Solution Submitted by Federico Kereki    
Rating: 3.0000 (5 votes)

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Solution Solution | Comment 7 of 25 |

Consider (x - 1)x(x + 1) = x(x - 1) = y.

Since the greatest common divisor of x and x - 1 is 1, we have x = a, x - 1 = b, for some natural numbers a and b.

But then (a) - b = 1, which is impossible if a and b are natural numbers.

Hence the product of three consecutive natural numbers cannot be a perfect square.

As a generalization, the product of any number of consecutive positive integers is never a perfect power.  This was proved by Erds and Selfridge in 1975.

  Posted by Nick Hobson on 2004-06-26 13:31:01
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