# Golden ratio / Fibonacci which branch of math?

Friends,

The Golden ratio / Fibonacci sequence are studied under which branch of math?

Can you recommend some good textbooks on the subject?

Thanks

• There is also "Fibonacci and Lucas Numbers, and the Golden Section - Theory and Application" by Steven Vajda. – Stiefel Aug 20 '14 at 9:01

## 3 Answers

I put my card in for Number Theory, only because there are, at minimum, cursory notes on this topic in every basic number theory book on my shelf. I have no number theory book to recommend at the moment (I like them all). There is however a really nice easy read on my shelf called The Divine Proportion by H.E. Huntley. It is a Dover publication so it should be pretty affordable. This would be my first recommendation for anyone with an interest in this particular topic (and no other lit.), and I consider it an especially good recommendation by virtue of the price, ease of read, and quality. It is not a textbook in the academia sense, but it should give you a good grasp of where to go and what the world finds interesting about the golden ratio and the Fibonacci sequence. While there are no textbook exercises, there is plenty of opportunity to break out the pencil and paper.

• I'm a bit surprised by Number Theory, because I expected counting/combinatorics or linear algebra. I tend to hear number theory address issues of primality / factoring / etc, what number theory results address the Fibo. seq? – DanielV Dec 8 '13 at 7:48
• In most elementary number theory books this arises as a section or chapter. For example, in a book I just pulled off the shelf by the title Elementary Number Theory by Kenneth H. Rosen, Fibonacci numbers come to topic in chapters titled "The Fibonacci Numbers", there are exercises in several chapters including one titled "Finite Continued Fractions", there is a proof that the Fibos grow faster than a geometric series, there is a demonstration for the explicit formula form for the Fibonaccis, and the list actually does continue. Many exercises related to the sequence. <character limit met> – J. W. Perry Dec 8 '13 at 8:01
• @DanielV In the same book, chapter "Pseudorandom Numbers" has several exercises on generating pseudorandom numbers using the "Fibonacci generator". These are elementary texts, but certainly number theorists belong in the crowd of people who have something profound to say about the golden ratio, Fibonacci sequence and the like. In the end, Fibonacci sequences are just numbers, so the number theoretic engagement should come as no surprise. Also consider the Fibonacci primes conjecture. Therein lies a parallel that may please you. :) I think Fibonacci was a number theorist. Read a bio <charac..> – J. W. Perry Dec 8 '13 at 8:45

Usually recurrence relations fall under the wide umbrella of discrete math. A nice book on the topic is Concrete Mathematics by Graham, Knuth, and Patashnik.

• Slightly less wide umbrella would be combinatorics. – Gerry Myerson Dec 8 '13 at 6:22

These topics, have their own chapter in K. K. Tung Topics in Mathematical Modeling. In fact, you can legally download that chapter for free because they offer the pdf of it as a sample on the book's website: http://press.princeton.edu/chapters/s8446.pdf .