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Show that there is no positive integer n, for which √(n-1) + √(n+1) is rational.

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Solution:
Let us assume that there is a positive integer n for which √(n-1) + √(n+1) is rational and equal to A/B, where A and B are positive integers
(B  0).
Then,

Since, A and B are positive integers. 
=> √(n-1) and √(n+1) are rationals. 
But it is possible only when n + 1 and n - 1 both are perfect squares. But they differ by 2 and any two perfect squares differ at least by 3. 
=> n + 1 and n - 1 cannot be perfect squares. Hence, there is no positive integer n for which √(n-1) + √(n+1) is rational.

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