You get unfamiliar with your tools if you don't use them. I'm sure there's fortune cookie wisdom in there somewhere. But avoiding this excellent excuse to write Haiku, I'll just state that if someone asks me about functional analysis, measure theory or reading some paper in French today, I'm sure to nod nostalgically. I might hold up a conversation about the subjects when I'm drunk, but I have no real working skill in graduate level mathematics anymore.
Anyway, Magic. This is still somehow related to the history of Magic. Very old school stuff though. Like before Alpha playtest cards old school. Like Richard Garfield January 1991 old school.
The distribution of the binomial coefficients modulo p. Hardcore stuff. I think I might understand the title, which is always a very good sign. Binomial coefficients are the family of integers that e.g. shows up when you want to pick some objects from a collection of said objects. Like if you want to pick three players for a deck-check in a tournament with 40 players, in how many ways can you do that? That's a binomial coefficient. Or if I want to calculate the odds of getting this particular FTK hand from my grossly unfair Power Monolith deck:
Modulo p then. Slightly simplified, the modulo operation typically finds the remainder after division of one integer by another. So for example 16 modulo 12 asks for the remainder of 16/12, i.e. 4. If two numbers give the same remainder when divided by a given number, they are said to be congruent. So 16 is congruent with 4 (and 28, 40, 52, etc) modulo 12.
There are many cases when modular arithmetic comes up in everyday life. The hour of the day is modulo 24. If it's 20:00 and you wait six hours, it's not 26:00 o'clock but rather 02:00. The months of the year is modulo 12; after December we get back to January again. So let's say that we would pick five different months since the start of the current calendar in England in 1752 (when they decided to start the year with January over there). That's 3169 months ago. What are the odds that all the five months we picked were October? If we pick five random numbers from 1 to 3169, each one of them need to be congruent with 10 modulo 12 to represent the tenth month of the year, and we can pick the months in 3169 choose 5 different ways. So far so easy.
Hm. Thinking about it, this could be the least user friendly topic even remotely related to the history of Magic one could write about. If I were writing on a commercial site, odds are that my editor would reject this with a vengeance. Well, I'm in too deep now. And lucky me, I have no editor. Here's a picture of Land Leeches:
So, the end result is used "to study how the the values of the binomial coefficients sit in the quadratic residues modulo p." What's a quadratic residue then? Lets look it up in my Introduction to modern algebra:
- 1² ≡ 1 mod 5
- 2² ≡ 4 mod 5
- 3² ≡ 4 mod 5
- 4² ≡ 1 mod 5
Lets take a quick step back and look at just how large numbers we're working with when talking about binomial coefficients. This is not related to the results in the thesis, but to get a better idea of what we would be dealing with without the modulo operation. In the introduction of the thesis, Garfield has a brief discussion where he uses n=10^9 as an example. That's about the size of the population of India in 1998. So say that you want to pick 1% of that population to do some survey on their living standards or similar. In how many ways could we pick out this one percent? I can't really write that binomial coefficient here, as it has well over 24 million digits. We would have to use all the characters of eight Bibles to write that number down. And if we go a little further, say picking 10% of the population rather than 1%, the binomial coefficient gets too big for most scientific calculators to handle. And we're still looking at, in a sense, small numbers here; 1 billion is a number we can relate to and picking 10% of that isn't that big a chunk.
So how do the values of binomial coefficients for different values of k sit in the residue groups modulo p? And why is it interesting? Well, paraphrasing Black Knight, math is its own purpose. It's the poetry of logical ideas. As Bertrand Russel so eloquently put it, "Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry."
I can't say I understand the proof. But dismissing its potential beauty because of my inept insight in the field would be ignorant. It's like someone calling from the Louvre to describe Mona Lisa on the phone and the respondent dismissing the painting as they wont get get a proper understanding of its nuances ("It's some chick with a weird smile and dark hair, with slightly tilted landscape in the background for flavor").
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