Question

Show that there is no positive integer n for which√n-1+√n+1is rational

Answer 1

Let 
√n+1+√n−1 is rational and can be expressed by pq
where p and q prime to each other and q≠0

So 
√n+1+√n−1=pq.........(1)

Inverting (1) we get

1√n+1+√n−1=qp

⇒√n+1−√n−1(√n+1+√n−1)(√n+1−√n−1)=qp

⇒√n+1−√n−12=qp

⇒(√n+1−√n−1)=2qp.....(2)

Adding (1) and (2) we get

2√n+1=pq+2qp

⇒√n+1=p2+2q22pq.....(3)

Similarly subtracting (2) from (1) we get

⇒√n−1=p2−2q22pq.....(4)

Since p and q are integers then eqution (3) and equation(4) reveal

that both √n+1and√n−1

are rational as their RHS rational

So both (n+1)and(n−1) will be perfect square

Their difference becomes (n+1)−(n−1)=2

But we know any two perfect square differ by at least by 3

Hence it can be inferred that there is no positive integer for which

√n+1+√n−1 is rational