Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Below, $x=\phi$ when $n=2$:


($\phi$ being the golden ratio)

Is there a way to express $x$ in terms of $\phi$ for $n>2$?

share|cite|improve this question
I hope you realize you can simplify that sum right ? I dont know if that helps much though. – mick Sep 2 '12 at 22:28
The positive roots $>1$ of $x^{n+1}-2x^n+1$.. looking at limiting ratios of successive terms defined by linear recurrence relations $a_{m+1}=a_m+\cdots+a_{m-n+1}$ are we? Relevant: – anon Sep 2 '12 at 22:37
What do you mean by "express"? – Qiaochu Yuan Sep 2 '12 at 22:40
up vote 2 down vote accepted

When $n=5$, we are talking about the roots of $x^5-x^4-x^3-x^2-x-1=0$. I suspect that, like most polynomials of degree 5, this one has Galois group $S_5$. If that's the case, then these roots can't be expressed in terms of the four arithmetical operations and square roots, cube roots, fifth roots, etc. It would follow that there's no expression in terms of the golden ratio (and arithmetic operations, and square roots, etc.).

It doesn't get any better for $n\gt5$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.