Prove that for any positive integer

. . is never a perfect square.

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- Jan 6th 2007, 11:10 AMSorobanQuickie #8
Prove that for any positive integer

. . is never a perfect square.

- Jan 6th 2007, 02:20 PMThePerfectHacker

Assume it is a square....

**Theorem:**If a product of a positive integer and a square is is square then that positive integer itself must be a square.

Thus,

Trivial-Improper factorization,

Thus,

.

But the problem says**positive integer**.... a contradiction.

This is mine 4:):):)th Post!!! - Jan 6th 2007, 04:39 PMgalactus
Here's a thought:

If you factor:

Take the square root:

If n is a positive integer, then k can not be an integer, therefore, can not be a perfect square. - Jan 6th 2007, 07:35 PMSoroban
Hello, Hacker and Galactus!

I like both your solutions.

The "Quickie" solution is surprisingly clever

. . but almost impossible to derive.

Let

Since

. . we have: .

Since

. . we have: .

Hence: .

. . and lies between two consecutive squares.