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May 29th, 2014, 12:14 PM   #1
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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.
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May 29th, 2014, 02:06 PM   #2
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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
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May 29th, 2014, 03:02 PM   #3
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$$x + \frac{1}{x} = n \qquad \rightarrow \qquad x^2 - nx + 1 = 0$$
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.
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May 29th, 2014, 04:10 PM   #4
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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.
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May 29th, 2014, 04:17 PM   #5
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Quote:
Originally Posted by JohnCoop View Post
Prob 23. For x - 1/x = 5 ... [show that] x^4 + 1/x^4 = 727
$\displaystyle x-\frac{1}{x}=5$

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

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

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

$\displaystyle x^4+\frac{1}{x^4}=727$
Thanks from Denis
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May 29th, 2014, 06:38 PM   #6
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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!
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May 29th, 2014, 07:04 PM   #7
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Quote:
Originally Posted by Denis View Post
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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