show that there is no positive integer n so that root(n-1)+root(n+1) is rational
Suppose there exists a positive integer n for which is a rational number.
, where p and q positive integers and q ≠ 0.
From (1) and (2),
⇒ n + 1 and n – 1 perfect square of positive integers.
Now, (n + 1) – (n – 1) = 2, which is not possible since any two perfect squares differ by atleast 3.
Hence, there is no positive integer n for which is a rational number.