During several teach-in todays, I witnessed a discussion by some mathematicians that worked in the military-industrial complex. One major thing of interest spoken of was that number theory, long considered useless, was now the basis of modern cryptography, a huge part of our lives today. I’ve heard this several times, so I’ve started wondering if at this point other pratical uses for number theory have been discovered. What’s the Straight Dope?
That part of Number Theory that pertains to permutation and symbolic relationships is needed big-time in the life sciences. The amount of data that has to be dealt with in “biological informatics” is absolutely monumental. Even a relatively simple phylogeny problem can take theoretically unlimited dedicated supercomputer time. Thus, number theory jockies are desperately needed to work on the nuts and bolts of these matters.
I’ve had a few jobs that exceeded the time allotment on massively parallel supercomputers. When I was recruited to test out a new parallel phylogeny application, the first thing I sent to it overwhelmed the poor program, and that was a limited version of the analysis I wanted to do.
IIRC the error correction algorithms used in music CD player electronics is based some discoveries related to number theory research.
Here are the following steps that are involved in the encoding process of digital music:
Number theory is used a lot in group theory, which is used pretty much everywhere.
The “hashing” used to index computer files.
An old mathematics department toast:
“To pure mathematics…May it never be of use to anyone!”
Reed-Solomon codes are part of coding theory. Various topics from combinatorics are needed to understand it, as well as finite field theory (and other parts of abstract algebra). I wouldn’t call it part of number theory. Hashing and error-correction are coding theory/combinatorics-related and not really part of number theory.
Number theory was the inspiration for group theory, and in that sense is used in it. Group theory was invented by taking ideas developed for number theory (and some other areas as well) and generalizing them. Number theory and group theory are taught in completely separate course these days. It’s possible to do group theory without knowing much number theory.
Dogface writes:
> That part of Number Theory that pertains to permutation and
> symbolic relationships is needed big-time in the life sciences.
I don’t know exactly what you mean by “permutation and symbolic relationships,” but I suspect that these are related to combinatorics, not number theory.
Number theory tends to be more an inspiration for other fields of mathematics than to be used in them. Fermat’s Last Theorem is a perfect example. The reason that it took 350 or so years to solve it was that it was necessary to create large amounts of other mathematical fields for the proof, fields which are useful for other things. These fields are not usually considered as part of number theory.
Very basic Number Theory is used all the time in Computer Science. Look at the Discrete Fourier Transform (DFT). It has a lot of applications. When working with a modular field, you need roots of unity in that field. Number Theory tells us what are good values to use.
Since modulo arithmetic is all over the place in CS, it’s basic properties are necessary to do things right. Primes that are one less than a power of 2 are quit popular for the modulus in many applications.
Quadratic fields are not so common, but they are used.
A really neat problem involves expander graphs: Linear number of edges but any bisection has to cut a large number of edges. They are used in designing what are basically switching networks. Some of the best techniques for constructing expander graphs are based on Number Theory.
And we are just getting started. You can assume that within a few decades every theorem in Hardy and Wright will be applied in CS. Good thing Hardy is dead, this would’ve killed him.