Math Newsletter: Fibonacci Numbers

Has anyone not heard of Fibonacci numbers? They’re found in nature, literature, movies, and well, they’re famous. They’re also on the Internet, so if you really want to delve into the subject, just go online. There, I imagine, you’ll get the official version. In this article, you’ll get mine. Some resemblance should be expected and would not be coincidental – after-all, all the characters “living or dead” are all dead.  Dead since the 11^th Century. So where did these numbers come from? Answer: rabbits; pairs of rabbits.

Fibonacci was tackling the problem of rabbit propagation. I can’t honestly say for sure whether he was interested in the problem ecologically or as a math-puzzle. Sources I’ve read are conflicting and Fibonacci possessed a middle-age mind – chronologically and historically. Regardless, here’s how we’ll proceed: (1) Rabbits go from baby pairs to adult pairs in one month, (2) an adult pair can conceive a baby pair after one month of adulthood (3) Rabbits live forever. All reasonable assumptions! 

The pattern that emerges is that of any three consecutive numbers, the last is the sum of the prior two. 

Or, speaking mathematically: F_n+1 = F_n-1 + F_n. The first 35 Fibonacci number are listed below:

Fibonacci Numbers

What makes these numbers so intriguing to mathematicians are the amazing patterns they possess. I have my favorites. The first can be Googled, but the proof is my own, the second I discovered myself, and the third – the most magical one, is my twist on well-known results.     

The first pattern is known as Cassini’s Identity. Look at any three consecutive Fibonacci numbers, for example, 13, 21 and 34.   Square the middle one (21² = 441) then multiply the outer two by each other (13 x 34 = 442). That 442 and 441 differ by one is no chance result – it always is the case.  Try any three pair yourself.

The second pattern has no formal name but is just as interesting. Identify the position of any Fibonacci number, say F₇ = 13, then look at the Fibonacci number in the doubled position – in this case F₁₄ = 377.

Turns-out that 13 divides 377. (377 = 13 x 29). Not only is this true for any doubled-position, it is true for any position multiple. That is, F₂₁ = 10946 = 13x 842 and F₂₈ = 317811 = 13x24447. Put another way, if a number divides the position of a Fibonacci number then the corresponding Fibonacci numbers divide as well. Let’s look at F₃₅  = 9227465. As 5 and 7 divide 35 then 5 (F₅ = 5) and 13 should divide 9,227,465 and sure enough they do, 9,227,465 = 5 x 13 x 141,961

But we’ve saved the best for last. Recall the Golden Ratio: the value of x where x/1 = (x + 1)/x. We talked about this in a previous newsletter where we found x to be 1.61803… Now pick two consecutive Fibonacci numbers, here big is better. Say F₃₁ and F₃₂, 1,346,269 and 2,178,309.  Divide the former into the later: 2,178,309 ÷ 1,346,269 = 1.61803…. Now if that’s not pulling a rabbit out of a hat, what is?

We wrap things up with a proof of our result. No harm in skipping if you’ve read enough already.

Here goes:

F_n

F_n+1   =   F_n  +  F_n-1

F_n+2   =   (F_n-1  +  F_n)  +  F_n  =  F_n-1 + 2F_n

F_n+3   =   (F_n-1 +  2F_n)   +  (F_n-1  +  F_n)    =  2F_n-1  + 3F_n

More generally, F_n+K  = F_k F_n-1 + F_k+1 F_n

If k = n, we have F_2n = F_n (F_n-1 + F_n+1), proving that F­_n  divides  F_2n

For F_{(a + 1)d}  =  F_{d + ad}   =  F_ad F_d-1  +  F_{ad + 1} F_d  -> if F_d divides F_ad then it divides F_{(a +1)d}

As F_d divides F_d , by mathematical induction we have our proof.

Going in reverse: 

F_n-2 = F_n  – F_n-1

F_{n- 3} = F_n-1  – F_n-2  =  F_n-1  – (F­_n  – F_n-1)  =  2F_n-1  – F_n  

F_n-4  =  F_n-2  – F_n-3   =  (F_n  – F_n-1) –  (2F_n-1  – F_n)  =  2F_n  – 3F_n-1

In the event k is even: F_n-k = F_n F_k-1  – F_k F_n-1

And in the event k is odd: F_n-k  = F_k F_n-1  – F_n F_k-1

Hence if k = n – 1 à 1 = F₁ = F_n F_n-2   +/– F_n-1 F_n-1 giving us Cassini’s formula.   

Our last result follows from the observation that F_n+1/F_n = (F_n + F_{n -1})/F_n  = 1 + (F_n-1/F _n)  = 1 + 1/(F_n/F_n-1).

Letting n go to infinity and assuming convergence (the proof of which is out of our league), we let x represent the limiting value giving us:  x = 1 + 1/x  = (x + 1)/x, the defining equation of the Golden Ratio.