* Don Reble <djr@nk.ca> [Apr 13. 2006 08:10]:
she asked what was the use of prime numbers? (besides being a fascination to math people)
Hardy and Koblitz say,
... both Gauss and lesser mathematicians may be justified in rejoicing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. -- G. H. Hardy, _A Mathematician's Apology_, 1940
G. H. Hardy would have been surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography (secret codes). -- Neal Koblitz, _A Course in Number Theory and Cryptography_, 1987
To me, it looks like Hardy is mostly right, even now. Computer-based communication seems to be the sole killer-application for number theory.
Will someone please disagree?
Me. Factorization into prime powers gives you the structure of the multiplicative group. Prerequisite to do basically anything. That we use computers instead of paper charts and hand computations makes no difference IMO. All FFT algorithms exploit the structure of the transform length. Rader's algorithm for prime length FFTs exploits the fact that a primitive root exists. Random number generators, and secure ones (BBS). Pseudo noise sequences via shift registers (for, e.g. spread spectrum). Watermarking with quadratic residues. Constructions for Hadamard matrices (shift registers, or, again quad.res.). Fast computations using CRT decomposition. [fill in 20 more here] Remove number theory from all electronic devices and nothing but flashlights would work a microsecond later. Just six month later there would probably be little left of what we consider a civilized world. Very very many people would die. I cannot see how the overwhelming importance of number theory (together with ECCs) should taint the underlying mathematics. I strongly dislike the attitude reflected by Hardy's words. Says jj, the computationalist P.S.: Gauss did a huge amount of computations.