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 wellknown 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: 1038 
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 wellknown; perhaps CRG is sleeping at the wheel 
May 29th, 2014, 03:02 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 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: 7,674 Thanks: 2654 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 
Senior Member Joined: Sep 2012 From: British Columbia, Canada Posts: 764 Thanks: 53  
May 29th, 2014, 06:38 PM  #6 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
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 
Senior Member Joined: Sep 2012 From: British Columbia, Canada Posts: 764 Thanks: 53  This is just $\displaystyle v=u^4+4u^2+2$. Quite uninteresting.
