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. 

  • 176
What are you looking for?