prove that there is no positive integer n for which root n - 1 + root n + 1 is rational.

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- May 18th 2010, 10:33 PMsaha.subhamreal number problem 3
prove that there is no positive integer n for which root n - 1 + root n + 1 is rational.

- May 18th 2010, 11:06 PMkompik
- May 18th 2010, 11:59 PMsaha.subham
its root n + 1 not whole root n + 1

- May 19th 2010, 12:07 AMkompik
- May 19th 2010, 07:17 AMsaha.subham
yeah exactly plzz solve it :)

- May 19th 2010, 07:32 AMkompik
is rational if and only is rational.

You get:

meaning that is rational and therefore (n-1)n is a perfect square.

Now since n-1 and n are coprime, both n-1 and n are perfect squares. So again you have two squares which differ by 1 and you can continue int he same way as I did in my first post. You get that the only solution is n=1.

BTW do you know the result, that a square root of an integer is rational if and only if this integer is a perfect square? If you did not learn this, you were probably supposed to look for a completely different solution. (Perhaps there is a simpler one and I have overlooked it...) - May 19th 2010, 09:48 PMsaha.subham

sorry i meant to said this one. extremely sorry ti disturb u again - May 19th 2010, 11:28 PMkompik