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First of all, I apologize for my amateurness and inexperience. Although I always enjoyed math, only two years ago I started experimenting with continued fractions and gained a deep reverence for them.

I first "discovered" the trivial but beautiful identities linking the golden ratio to the n-th Lucas number through raw experimentation:

$φ^n = L_n-\cfrac{1}{L_n-\cfrac{1}{L_n-\cfrac{1}{L_n-\cdots}}} $ $, n∈2N$

$φ^n = L_n+\cfrac{1}{L_n+\cfrac{1}{L_n+\cfrac{1}{L_n+\cdots}}}$ $, n∈2N+1$

$\lim_{n\to\infty} ϕ^n=L_n$

Which follows from $\phi^n + \phi^{-n} = L_n$ $, n∈2N$

$\phi^n - \phi^{-n} = L_n$ $, n∈2N+1$

I gained great enjoyment from messing about with numbers but unfortunately starting university had distracted me until recently, when right before I slept I somehow jotted down the following trivial but "beautiful" results:

$e = 3-\cfrac{1}{3+\cfrac{1}{2-\cfrac{1}{5+\cfrac{1}{2-\cfrac{1}{7+\cfrac{1}{2-\cfrac{1}{9+\cdots}}}}}}} $

$e = 1+\cfrac{1}{1-\cfrac{1}{2+\cfrac{1}{3-\cfrac{1}{2+\cfrac{1}{5-\cfrac{1}{2+\cfrac{1}{7-\cdots}}}}}}} $

While I realize all of these results have been probably published hundreds of years ago, the process of independently finding them out is extremely enjoyable to me. I have become obsessed with mathematics! I love it and often spend so much time reading and experimenting that I fall asleep exhausted. I would like to progress further with mathematics. Would any of the more experienced members have any recommended introductory texts on the topic of continued fractions and mathematics in general? Thank you for your patience and time reading this.

Mohamed Sabba
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  • This is a pretty good one: http://smile.amazon.com/gp/product/0486696308?psc=1&redirect=true&ref_=oh_aui_search_detailpage – Gregory Grant Feb 04 '16 at 00:22
  • The C. D. Olds Continued Fractions is a classic. Out of print, but maybe you can find a cheap used copy. It is very easy reading as math books go. – J. W. Perry Feb 04 '16 at 00:31
  • http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/neverending-fractions-introduction-continued-fractions – Will Jagy Feb 04 '16 at 01:30
  • http://www.amazon.com/Introduction-theory-numbers-Leonard-Dickson/dp/B0007DQ0OK – Will Jagy Feb 04 '16 at 01:37
  • Thank you for your recommendations! I should be receiving the book by Khinchin today, and have the Olds and Dickson .pdfs downloaded. I will be looking to purchase the Borwein book in the very near future. What a shame these books are not available at my campus. – Mohamed Sabba Feb 05 '16 at 14:06
  • The reason I included Dickson is that some typical uses of continued fractions are in number theory. When $\gcd(a,b)=1,$ they are the best way to write the calculation of $ax+by=1.$ Next, Lagrange's method for solving $x^2 - n y^2 = 1$ with $y \neq 0$ is a better version of CF's, the chapter about chains of indefinite binary quadratic forms. Or, for that matter, $ax^2 + b xy + c y^2 = k,$ when $|k| < \sqrt \Delta.$ Here $\Delta = b^2 - 4 a c$ – Will Jagy Feb 05 '16 at 16:02

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