Is $sqrt(n+1)+sqrt(n-1)$ rational or irrational ?

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Answer 1 · 2016-09-23T20:21:47

See below.

We will prove that, searching for the rational solutions.

Supposing that $sqrt(n+1)+sqrt(n-1)$ is rational then

$sqrt(n+1)+sqrt(n-1)=p_1/q_1$ so

$(sqrt(n+1)-sqrt(n-1))(sqrt(n+1)+sqrt(n-1))=n+1-(n-1)=2 = (sqrt(n+1)-sqrt(n-1))p_1/q_2$ is also rational

So $sqrt(n+1)-sqrt(n-1) = p_2/q_2$ is also rational then

$2sqrt(n+1) = p_1/q_1+p_2/q_2 = (p_1q_1+p_2q_2)/(q_1q_2)$

So $sqrt(n+1) = 1/2 (p_1q_1+p_2q_2)/(q_1q_2)$

With the same argument we could prove also that

$sqrt(n-1)$ is rational.

So $sqrt(n^2-1) = p/q$ is rational and consequently

$n^2 = (p/q)^2+1$ Supposing that $p/q = m$ ( remember that $n$ is integer) we will need

$n^2=m^2+1$ or

$(n+m)(n-m) =1$ with $n,m$ integers, $m = 0, n= 1$ being the only rational solution.

Concluding, the only rational solution is for $m = 0, n=1$ but then

$sqrt2$ is irrational so for $n in NN^+$ , $sqrt(n+1)+sqrt(n-1)$ is irrational.

Is sqrt(n+1)+sqrt(n-1)sqrt(n+1)+sqrt(n-1) rational or irrational ?