The context of the student's question made it seem like primes were "invented" in order to solve some particular problem. However, primes drop out naturally from the basic axioms of arithmetic (Peano), so they are there whether you want them or not. Since most people find the axioms of arithmetic pretty compelling, they get primes "for free". A crude analogy is the "two party" political system in the U.S. For better or for ill, the various constraints of the U.S. Constitution seem to force a two party solution. Given the number of challenges to this two party system over a long period of time, the continued dominance of the two parties indicates a remarkably stable equilibrium. The fact that this does not happen with other constitutional systems seems to indicate that some unique combination of features in the U.S. version conspire to produce this equilibrium. Of course, non-linear systems can have incredibly complex behaviors, with many different equilibrium points (both stable & non-stable), so we may yet see different behaviors from this system. Mathematicians analyze systems like these for their basic properties, while engineers try to take advantage of these properties in order to accomplish some particular goal. Mathematicians would continue to study primes forever, regardless of their "usefulness"; engineers would continue to attempt to "use" primes to build something. At 10:30 PM 4/12/2006, mcintosh@servidor.unam.mx wrote:
Quoting "Stan E. Isaacs" <stan@isaacs.com>:
I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. ....
I was going to mention the incommensurable breeding cycle of locusts but someone beat me to it. But how about turning the question around and asking about what good are non-primes? Suppose we had 23 hour days, or sold bread in baker's dozens, or (heaven forbid) tried to use the metric system. There wouldn't be any half-pints of beer, or packaging eggs in rectangular crates or doleing out pizza slices at a kids' birthday party.
By knowing where the primes lie, we can avoid all this anguish.
- hvm
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