I‘m trying to understand a proof on my textbook that says $sqrt {n − 1} +sqrt {n + 1}$ is irrational, where $n in mathbb{N} $. However I can’t seem to show that

$$n^2 – 1 = frac{(frac{p^2}{q^2} – 2n)^2}{4} to n = frac{(p^4+4q^4)}{4p^2q^2}$$,

where $p,q in mathbb{Z}$，and the gcd of $p,q$ is 1. I feel really stupid, is there something I don’t see?

The furthest I got is

$$n = frac{sqrt{2p^4+5p^4-2p^2q^2}-(p^2-q^2)}{2q^2}$$

Mathematics Asked by Heng Wei on December 30, 2020

2 Answers$$n^2 - 1 = frac{left(dfrac{p^2}{q^2} - 2nright)^2}{4}= frac{dfrac{p^4}{q^4} + 4n^2-4ndfrac{p^2}{q^2} }{4}=dfrac {p^4}{4q^4}+n^2-ndfrac{p^2}{q^2}. $$

Subtract $n^2$ from both sides to get $$-1=dfrac {p^4}{4q^4} -ndfrac{p^2}{q^2} ,$$

which is $$ndfrac{p^2}{q^2}=1+dfrac {p^4}{4q^4}.$$

Can you take it from here (solve for $n$)?

Correct answer by J. W. Tanner on December 30, 2020

If $$n^2 - 1 = frac{(frac{p^2}{q^2} - 2n)^2}{4}$$ then $$n^2 = frac{(frac{p^2}{q^2} - 2n)^2+4q^2}{4}$$ then $$n^2 = frac{(p^2 - 2q^2n)^2+4q^2}{4q^2}$$ then $$4q^2n^2 = (p^2 - 2q^2n)^2+4q^2$$ then $$4q^2n^2 = 4q^4n^2-4p^2q^2n+p^4+4q^2$$ so $$0=-4p^2q^2n+p^4+4q^2$$ and finally $$n=frac{p^4+4q^4}{4p^2q^2}$$

Answered by Measure me on December 30, 2020

1 Asked on January 7, 2022

4 Asked on January 7, 2022

3 Asked on January 7, 2022

2 Asked on January 7, 2022 by mondo-duke

1 Asked on January 7, 2022

abstract algebra finite groups group actions group theory holomorph

0 Asked on January 7, 2022

1 Asked on January 7, 2022

41 Asked on January 7, 2022

1 Asked on January 7, 2022 by roslavets

3 Asked on January 7, 2022 by as-soon-as-possible

1 Asked on January 7, 2022

arithmetic functions divisor sum number theory ordinary differential equations perfect numbers

1 Asked on January 7, 2022 by christopher-rose

lebesgue integral measure theory real analysis sequences and series

1 Asked on January 7, 2022

continuity distribution theory measure theory sobolev spaces topological vector spaces

4 Asked on January 7, 2022 by dcdaking

1 Asked on January 7, 2022 by brucemcmc

1 Asked on January 7, 2022 by fred-akalin

differential geometry manifolds smooth manifolds taylor expansion

2 Asked on January 7, 2022 by hwood87

2 Asked on January 7, 2022

1 Asked on January 7, 2022 by zhanxiong

Get help from others!

Recent Questions

- Why fry rice before boiling?
- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?

Recent Answers

- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved.