Let
sqrt(n+1)+sqrt(n-1)" is rational and can be expressed by " p/q
where p and q prime to each other and" "q!=0
So
sqrt(n+1)+sqrt(n-1)=p/q.........(1)
Inverting (1) we get
1/(sqrt(n+1)+sqrt(n-1))=q/p
=>(sqrt(n+1)-sqrt(n-1))/((sqrt(n+1)+sqrt(n-1))(sqrt(n+1)-sqrt(n-1)))=q/p
=>(sqrt(n+1)-sqrt(n-1))/2=q/p
=>(sqrt(n+1)-sqrt(n-1))=(2q)/p.....(2)
Adding (1) and (2) we get
2sqrt(n+1)=p/q+(2q)/p
=>sqrt(n+1)=(p^2+2q^2)/(2pq).....(3)
Similarly subtracting (2) from (1) we get
=>sqrt(n-1)=(p^2-2q^2)/(2pq).....(4)
Since p and q are integers then eqution (3) and equation(4) reveal
that both" "sqrt(n+1) and sqrt(n-1)
color (blue)(" 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
sqrt(n+1)+sqrt(n-1)" is rational"
