Stan E. Isaacs wrote
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. It caught me unawares, I mumbled something about that there are theorems where it is easier to prove them for primes, and then extend the proof to all numbers, but I couldn't think of an example. I looked around a little, but most of the things I see, off-hand, talk about all the things you can prove about primes themselves, not what use they might be elsewhere. I could probably mention mods, since you need mod p to be prime, to allow division. But what I really want is some examples where they might be of use in "everyday life", or at least in some mathematics that would apply to more everyday things. Anyone have ideas of what to tell her? Or references of places I could look?
I suggest you take a look at the sixth section of Carl Friedrich Gauss' Disquisitiones Arithmeticae http://www.math.uni-bielefeld.de/~sieben/DA_sechster_abschnitt.pdf As an example Gauss calculates the decimal development of the fraction 6099380351/1271808720 (a fraction coming up as an approximation of \sqrt{23}) to as many digits as one wants by a method he invented to avoid long division. Prime numbers, `mods' and the `unique decomposition of a number into a product of primes' enter as efficient tools in a task that everyone knows from elementary school. Hereby Gauss' makes use of a table with the periods of the decimal fractions of the inverses of all primes and prime powers less than 100 at the end of the Disquisitiones which is part of part of the much bigger table http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D137285 which he finished to compute the age of 18 -- the day before he left Brunswick to become a student in Goettingen. Best regards, Christian. -- Christian Siebeneicher e-mail:sieben@math.uni-bielefeld.de Universitaet Bielefeld http://www.math.uni-bielefeld.de/~sieben Fakultaet fuer Mathematik Postfach 10 01 31 tel secretary: (49) 0521-106 5022 D-33 501 Bielefeld fax: (49) 0521-106 6482 Germany tel home: (49) 0521-105335 fax home: (49) 0521-105325