MHB Rational Number: Proving $x+\dfrac{1}{x}$ is Rational

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Rational
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Let $x$ be a non-zero number such that $x^4+\dfrac{1}{x^4}$ and $x^5+\dfrac{1}{x^5}$ are both rational numbers. Prove that $x+\dfrac{1}{x}$ is a rational number.
 
Mathematics news on Phys.org
For each integer $n$, let $R_n = x^n + \dfrac1{x^n}$. If $m>n$ then $$R_mR_n = \left( x^m + \dfrac1{x^m}\right)\left( x^n + \dfrac1{x^n}\right) = x^{m+n} + x^{m-n} + \dfrac1{x^{m-n}} + \dfrac1{x^{m+n}} = R_{m+n} + R_{m-n}.$$ In the case when $m=n$ that becomes $R_{2n} = x^{2n} + \dfrac1{x^{2n}} = \left( x^n + \dfrac1{x^n}\right)^2 - 2 = R_n^2 - 2.$

Given that $R_4$ and $R_5$ are rational, it follows that $R_8$ and $R_{10}$ are rational.

Next, $R_8R_2 = R_{10} + R_6$ and $R_4R_2 = R_6 + R_2$. So $R_8R_2 = R_{10} + (R_4-1)R_2$ and therefore $R_2 = \dfrac{R_{10}}{R_8 - R_4 + 1}$. Hence $R_2$ is rational, and so is $R_6 = (R_4-1)R_2$.

Finally, $R_5R_1 = R_6 + R_4$, so that $R_1 = \dfrac{R_6+R_4}{R_5}$, which is rational.
 
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Back
Top