My Math Forum x +/- 1/x = n

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 May 29th, 2014, 12:14 PM #1 Newbie   Joined: May 2014 From: USA Posts: 1 Thanks: 0 x +/- 1/x = n A decade ago, I was tutoring a gifted grammar school student in Florida when she came upon the following problem in a Florida state problem set Prob 23. For x - 1/x = 5 Show its positive root is x = 5.192582 ... and x^4 + 1/x^4 = 727 exactly. I wondered whether this result could be generalized and now, a decade later, a colleague and I have found that for the positive root of x - 1/x = n .. ==> .. x^m + (-1/x)^m = N and for the positive root of x + 1/x = n .. ==> .. x^m + (1/x)^m = N where n, m and N are integers. I would like to know whether this result is well-known and/or has a name that would enable me to learn what further is known about it.
 May 29th, 2014, 02:06 PM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039 u = x - 1/x x = [SQRT(u^2 + 4) + u] / 2 v = x^4 + 1/x^4 Results (1st 10): u,v 1,7 2,34 3,119 4,322 5,727 (your example) 6,1442 7,2599 8,4354 9,6887 10,10402 Sloane's encyclopedia of integer sequences is unaware of that sequence, which indicates not well-known; perhaps CRG is sleeping at the wheel
 May 29th, 2014, 03:02 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra $$x + \frac{1}{x} = n \qquad \rightarrow \qquad x^2 - nx + 1 = 0$$ So the equation clearly has two roots. Moreover, by inspection they are clearly $a$ and $\frac{1}{a}$. Also $$x^2 + 2 + \frac{1}{x^2} = n \qquad \Rightarrow \qquad x^2 + \frac{1}{x^2} = n_2 \in \mathbb{Z}$$ Writing $u = x^2$, we easliy see that $$x^4 + \frac{1}{x^4} = N \in \mathbb{Z}$$. So that part can't be new. And that covers $m = 2^k$. I'd need to write more down to get as general as the OP, but I think it's unlikely to be new. Indeed, someone posted a question like this only a few weeks ago (for the case that I've just gone through I think). It's an interesting result though.
 May 29th, 2014, 04:10 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra More time turned out to be about 10 minutes. They are easy proofs by induction. I'd be amazed if it wasn't widly known. It's a nice spot though.
May 29th, 2014, 04:17 PM   #5
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Quote:
 Originally Posted by JohnCoop Prob 23. For x - 1/x = 5 ... [show that] x^4 + 1/x^4 = 727
$\displaystyle x-\frac{1}{x}=5$

$\displaystyle x^2-2+\frac{1}{x^2}=25$

$\displaystyle x^2+\frac{1}{x^2}=27$

$\displaystyle x^4+2+\frac{1}{x^4}=729$

$\displaystyle x^4+\frac{1}{x^4}=727$

 May 29th, 2014, 06:38 PM #6 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039 Same thing for x^2 - 1/x^2; 1st 10: 3,6,11,18,27,38,51,66,83,102 CRG s'got that one!
May 29th, 2014, 07:04 PM   #7
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Quote:
 Originally Posted by Denis Results (1st 10): u,v 1,7 2,34 3,119 4,322 5,727 (your example) 6,1442 7,2599 8,4354 9,6887 10,10402 Sloane's encyclopedia of integer sequences is unaware of that sequence, which indicates not well-known; perhaps CRG is sleeping at the wheel
This is just $\displaystyle v=u^4+4u^2+2$. Quite uninteresting.

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