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 Multiplying gets a square? (Posted on 2004-06-26)
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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 re(2): Solution | Comment 9 of 25 |
(In reply to re: Solution by Richard)

Yes, that is more direct.

However, consider (x - 1)x(x + 1) = x(x² - 1) = yn.

Since the greatest common divisor of x and x² - 1 is 1, we have x = an, x² - 1 = bn, for some natural numbers a and b.

But then (a²)n - bn = 1, which is impossible if a² and b are natural numbers.

And, using the same approach, we have proved the more general result that no product of three consecutive natural numbers can be a perfect power!

 Posted by Nick Hobson on 2004-06-26 18:53:02
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