White Rabbit Math - Extended

***I first wrote most of this post in 2008, but today an event reminded me of it, and so I thought I would add on to this old, but still interesting post with an additional interesting connection.

One of the things that amazes me, and I think most people who are attracted to math, is the mysterious way that different parts of math come together in unexpected ways. I tried to explain this to someone once using a literary analogy..."It is as if you were reading along in some great drama, or trying to understand the message in some grand poem, and suddenly the White Rabbit from Alice in Wonderland comes running through muttering, "Oh dear! Oh dear! I shall be too late!"
It is not the White Rabbit you see in math, but the effect is the same. Euler must have felt that feeling after he struggled to find the value of the series \( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2}+ ...\).. and finds that it turns out to be \( \frac{\pi^2}{6} \). Wait.... Pi is the ratio of the circumference to the diameter of a circle, but there are no circles in the sum of the squares of the reciprocals of the integers; and yet, there it is, the mathematical white rabbit coming seemingly from nowhere. Certainly none of the many mathematicians of great repute who had worked on the problem found (or expected) Pi to appear.

The normal distribution is another example; De Moivre takes the binomial probability distribution for flipping a coin and generalizes it toward an infinite number of flips, and POW, the normal or bell-shaped curve that is ubiquitous in intro stats. And what happens? Right there in the middle, the height of the normal curve at Z=0 is .39894... No, NO, NO, NOT JUST .39894.. but the .39894... that is exactly equal to \( \frac{1}{\sqrt{2 \pi}} \)

Ok, so what brought this sudden rebirth of excitement about mathematical interrelationships? Well recently I came across a blog that referred to another blog that (as these things sometimes do) led me to a paper on just such a mathematical "white rabbit". The paper was about partitions of numbers as powers of two (1, 2, 4, 8, 16, etc..)
It began with a simple question, what is the number of ways to write a number n as a sum of powers of two if each value can be expressed no more than two times. For example, we could express 4 as 4, or as 2+2, or as 2 + 1 + 1 since each value is a power of two, and none appears more than twice. You couldn't use 1+1+1+1 since it appears more than twice. For n= 4 it turns out that the number of partitions, as shown above, is three. If we assume that there is one way to express zero, and one way to express one, and figure out the others we get a string like this


1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7,..
Ok, you don't see a white rabbit yet... but then someone ask you a different question. Is it possible to write out ALL the rational numbers in simplified form without repeating any of them. The answer is "Yes, of course, see the list above."
"What?", you ask, "How?", but there it is... The sequence of rational numbers is formed by taking each of the numbers to be the numerator, and using the number behind it to be the denominator. 1/1; 1/2; 2/1; 1/3; 3/2; ... and you never get a repeat, never get an unsimplified form, and you eventually get them ALL, the entire Infinite Set.....
No way you would expect that partitions of powers of two should give you the rational numbers in their entirety... there is (it would seem) nothing to relate the two questions... and yet... there it is. I think that is what makes math the most exciting area of study in the world.
Prove it you say? Nope, In truth I ain't man enough, but you can find the entire paper
Recounting the rationals, by Neil Calkin and Herb Wilf. Read their proof and Enjoy.

*** So today I was catching up on some old audio podcasts from "My Favorite Theorem," and Jordan Ellenberg   was explaining his choice of a special part of Fermat's Little Theorem, that for any prime p, \( 2^p \equiv 2 Mod p \).   (or in very primitive terms, if you divide 2^p by p, you always get a remainder of 2.  I wondered why he found that so interesting, but then he hit me with, "you can discover at least that it’s true on your own, for instance by messing with Pascal’s Triangle, for example." And of course, in a moment I realized yes, Fermat's Little Theorem, at least this limited case, is elementary true by looking at the rows of Pascal's Triangle. The sum of all the elements of any row add up to a power of two, and the pth row has a sum of 2^p. But look at some prime row.....

the 3rd has 1,3,3,1 ;

the fifth has 1,5,10,10,5, 1 ;

and the 7th has 1, 7, 21, 35, 35, 21, 7, 1....

In each row, all the entries are divisible by p, except the two ones. Scan the rest and you notice the same thing. And just importantly, you don't have to go very far to see an exception for the non-primes.

Math has those White Rabbits everywhere.