Fibonacci Would Blush

Phantastic Phi

Sat, 27 May 2006 01:39:32 GMT

The Golden Number. (1 + √5) / 2. Phi. φ. Perhaps one of the coolest numbers around.

Let's take some powers of φ.

φ² = φ * φ
= [(1 + √5) / 2] * [(1 + √5) / 2]
= [1 + √5]² / [2]²
= [1 + 2*√5 + 5] / 4
= [6 + 2*√5] / 4
= (4 / 4) + (2 / 4) + (2*√5) / 4
= 1 + (1 / 2) + (√5)/2
= 1 + (1 + √5) / 2
= 1 + φ

φ³ = φ² * φ
= [1 + φ] * φ
= φ + φ²
= φ + [1 + φ]
= 1 + 2φ

φ⁴ = φ * φ³
= φ * [1 + 2φ]
= φ + 2φ²
= φ + 2 * [1 + φ]
= φ + 2 + 2φ
= 2 + 3φ

φ⁵ = φ * φ⁴
= φ * [2 + 3φ]
= 2φ + 3φ²
= 2φ + 3 * [1 + φ]
= 2φ + 3 + 3φ
= 3 + 5φ

φ¹ = 0 + 1φ
φ² = 1 + 1φ
φ³ = 1 + 2φ
φ⁴ = 2 + 3φ
φ⁵ = 3 + 5φ

And if you think that there's a pattern there, you're absolutely right. The Fibonacci sequence - clear as day. If you were to predict that φ⁶ = 5 + 8φ you would be absolutely correct.

Now I know there is a closed formula to find any particular number of the Fibonacci sequence (that is to say, if you enter any number n the function will give you the n^th term in the Fibonacci sequence. I know this exists because I have it written down somewhere in my discrete math notebook.

Unfortunately, said notebook is buried in a precarious three foot tower of notebooks. However, last week at work I decided to use the powers of φ to derive some sort of formula.

Unfortunately (again), I did not arrive at the simple closed formula that I have written down elsewhere. However, I did indeed come up with something that will produce any number of the Fibonacci sequence - although it is much less eloquent.

n
∑ (-1)^{(i + 1)} * [φⁿ / φ^{(2i - 1)}] = F(n)
1

Like I said, perhaps not the most eloquent little formula in the world - but it totally works. Actually deriving it was a blast. I wish I could say more, but there's only so much can do with ASCII (or rather so much that I'm willing to do) and my notes just aren't transcribable.

Though it's no real coincidence that the Fibonacci sequence shows up when fiddling around with φ. After all, if you take two sequential numbers in the Fibonacci sequence and divide the larger by the smaller, their quotient approaches the funny little value of 1.61803398874....

Humility

Tue, 07 Feb 2006 04:39:35 GMT

No, not a 2WW enchantment.

Instead, something completely simple absolutely stumped me*.

We all know that to divide one fraction by another fraction, you invert the denominator and multiply it by the numerator. Simple algorithm of the pre-algebra level. Then the following was presented:

Create a real world problem where you would have to divide two thirds by four fifths. That is to say...

2/3
4/5

I will admit, I couldn't answer it - I had some inklings of ideas, but ultimately was defeated. And you know, I'm a little glad that I was. I think as a result of that I appreciate the course much more than I would have if I came up with an example lickedy split.

* Apparently Dr. Halloran proposed the same question to a group of people with their doctorates in mathematics and elementary math teachers. It took an hour for them to come up with a solution, working in teams. It was an elementary school teacher who was the first to solve it.

Magnificent Mersenne Mathematomancing

Tue, 10 Jan 2006 02:39:36 GMT

In [much delayed] news, the 43^rd Mersenne Prime was found last month, on December 15, 2005.

What is a Mersenne Prime? A prime number that can be written by the expression 2^p - 1 where p is a prime.

Pretty simple. Pretty cool. It's a neat little trend, working with p = 2, 3, 5, 7, 13, 17, & 19 however 11 is sadly missing from that list (2¹¹ - 1 = 2,047 = 23 * 89). And, as my first line mentions, there have only been 43 discovered to date.

This latest prime is a number that is 9,152,052 digits long (p = 30,402,457). Not only that, but it is the largest known prime number to date. I thought that was particularly exciting, as it serves to give some added importance to this seemingly sparse method of finding primes.

And if it strikes your fancy, check out a bit more: http://www.mersenne.org/

Yeah, that's right

Wed, 30 Nov 2005 06:09:10 GMT

maltlick: Your mother is so fat that recurvise function used to calculate her obseity caused a stack overflow.

So good.

If one spiral isn't good enough....

Wed, 23 Nov 2005 04:30:37 GMT

http://www.numberspiral.com/

Totally interesting and most definitely on the stack for things to look at much more in depth once a few of these school effects resolve. I'm really sad that the Downloads link doesn't lead to anything.

Fibonacci and Phi Fervor

Wed, 16 Nov 2005 20:10:39 GMT

Tease
Holy shit! There's a closed formula for the Fibonacci sequence!

Foreplay
φ = (1 + 5^½) / 2
φ* = (1 - 5^½) / 2

(φ*, if I'm not mistaken, is the conjugate of φ. Please feel free to correct my terminology.)

Climax
F_n = (φⁿ - φ*ⁿ) / 5^½

Pillow Talk
{F_n} = {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...}

-----

When first blurted out over sushi harsh jugement was received. Well screw them, that's not enough to bring me down. I thought that this was absolutely fantastic because I hadn't know it existed. I did out a handful of calculations longhand, just to see how it worked at Video Time (meaning I don't have those notes on me). I had wrongly assumed that actually having 5^½ wasn't important, and any X^½ would have done. That, however, was incorrect.

It was interesting to see how the pattern (that is, element X being the sum of the previous two elements) showed up in the algebra (but with different numbers - R, S, T, U, V ...* as opposed to 1, 2, 3, 5, 8, ...). Anyhow, powers of φ are pretty neat.

*R, S, T, U, V will eventually be numbers, after I goto work later this week

Completely off topic

Tue, 18 Oct 2005 15:23:37 GMT

Query: mathematomancer or mathematemancer ?

I spent a couple hours stewing over that [albeit minor] choice before this journal's inception. On the one hand, I liked the e because it mimicked mathematics. On the other hand, the o was consistent with, say, necromancer.

Thoughts are appreciated.

Perilous Prime Programing

Tue, 11 Oct 2005 20:15:03 GMT

Before Discrete today, I wrote this program to [relatively quickly] generate a list of primes on the T1-83 calculator.

-----
Description: Outputs all prime numbers (into L1) from 3 to X (user defined).
Program: PRIME

1→Y:{3}→L1

Input "X: ",X

For(A,3,X,2)
1→Z
A^(1/2)→S

For(B,1,Y,1)
L1(B)→D

If int(A/D)=(A/D):Then
Y→B:0→Z:End

If D>S:Then
Y→B:End

End

If Z=1:Then
Y+1→Y:A→L1(Y):End

End
-----

The program isn't perfect, because in order to work it starts with 3 as the first prime (instead of 2). I also hate starting something off with an assumption as opposed to begining the program with an absolutely clean slate. Also for the sake of efficiency it only tests odd numbers instead of every integer (hence the first For statement increasing by 2 as opposed to 1).

Y is the # of elements in L1 (i.e. the # of found primes).
Z is the flag which denotes if a number is prime or note (1 means prime, 0 means composite).
S (and the statement D>S) is used to speed up the program and not test for values which are larger than the square root of the number.
L1 could start as 2 if the first For statement read "For(A, 3, X, 1)" however then it would check all even numbers and decrease efficiency