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Sum Root Concern (Posted on 2015-03-14) Difficulty: 3 of 5
Does there exist an integer n such that:
√(n-1) + √(n+1) is a rational number?

If so, provide an example.
If not, prove the non-existance of such an integer.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Computer solution (spoiler) Comment 1 of 1
Just kidding.  No computer was harmed in the solving of this problem.

If sqrt(n-1)+sqrt(n+1) is rational, then so is its square.

Squaring gives 2n+2*sqrt(n^2-1), and this is clearly rational only if n = 1.

So n = 1 is the only possibility, but this possibility does not work, 
because sqrt(0) + sqrt(2) is irrational.

So there is no such n.

Happy Pi-Day!

  Posted by Steve Herman on 2015-03-14 10:12:41
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