Fabulous Fibonacci

I started reading about perfect numbers (see previous post) because I ran into them in yet another book that I started over the weekend - The Fabulous Fibonacci Numbers by Alfred S. Posamentier and Ingmar Lehmann.

Of course most of us have heard about Fibonacci numbers that carry the name of the famous Italian mathematician, Leonardo of Pisa aka Leonardo Fibonacci (c. 1170 – c. 1250), who brought Arabic and Hindu numerals to the western world in his famous Book of Calculation, the Liber Abaci. My first introduction to them was perhaps through having to write my first algorithm and a computer code (in Fortran) for producing these numbers. But beyond that I rarely encountered them in my studies or reading and was astounded (yes..astounded) at some of the fascinating properties these numbers have. Transcribed below are some of the interesting ones I have read about so far.

1. Two consecutive Fibonacci numbers do not have common factors, i.e. they are ‘relatively prime’

2. F_n is composite (not a prime), if n is composite.

So, does that mean, Fn = prime, if n = prime? Not so..though amazing how many times it is true.

F₂ = 1 , not considered prime

F₃ = 2 = prime

F₅ = 5 = prime

F₇ = 13 = prime

F₁₁ = 89 = prime

F₁₃ = 233 = prime

F₁₉ = 4181 = 37 x 113, not prime

F₂₃ = 28657 = prime

…

10.
F₃, F₆, F₉, F₁₂, F₁₅, F₁₈…are all even and divisible by 2 or F₃

F₄, F₈, F₁₂, F₁₆, F₂₀, F₂₄… are all divisible by 3 or F₄

F₅, F₁₀, F₁₅, F₂₀,… are all divisible by 5 or F₅

And so on…

i.e. a Fibonacci number F_nm is divisible by _Fm

or in other words, if p is divisible by q, F_p is divisible by F_q.

15. F_n / F_{n+1 ~} golden ratio, phi = 0.618….a

F_n / F_n+2 ~ 0.38197….b

F_n / F_n+3 à 0.23606 …c

Note: a + b = 1.0 and b + c = a = phi!

More about the golden ratio, phi, later.

16. Natural number n can be written as an ordered sum of ones and twos in F_n+1 ways.

Eg: 5 can be written as 1+1+1+1+1

1+1+1+2

1+1+2+1

1+2+1+1

2+1+1+1

1+2+2

2+1+2

2+2+1 …. In 8 = F₆ ways.

Two items that show an interesting relation between Fibonacci numbers and Pythagoras triples:

17. Take any four consecutive Fibonacci numbers

- Product of middle two #s and double it = x

- Multiply outer two #s = y

- Add squares of inner two #s = z

Then, x, y and z form a Pythagoras triple i.e. x² + y² = z²

18. Babylonians had the following formula for obtaining Pythagoras triples (a, b, c)

a = m² - n²

b = 2mn

c = m² + n²

where m and n are natural numbers.

(a, b, c) are called primitive, if abc are relatively prime i.e. there are no common factors between a, b, c. This occurs when m and n are two relatively prime, with m > n and of different parity i.e. one is odd and the other even.

If m and n take on values of Fibonacci numbers, then C is a Fibonacci number too.