Talk:Computational number theory

I came to this page after linking to it from Peter Montgomery and was rather distressed with what I found. This stub is not only poorly written, it is not even an accurate stub. Other than mentioning primality tests, it does not describe the field of computational number theory at all, it describes algorithms for performing basic arithmetic operations that are important in many fields and already discussed on other pages (e.g. multiplication algorithms, modular exponentiation, Euclidean algorithm). Such algorithms are certainly used by computational number theorists, but they are not the main focus of computational number theory (for example, the finite element method is used by researchers in many fields, including quantum mechanics, continuum mechanics, and condensed matter physics, for example, but you don't find a description of it on the pages describing these fields)

A starting point might be the Princeton Companion to Mathematics article by Carl Pomerance https://math.dartmouth.edu/~carlp/PDF/pcm0049.pdf, which discusses primality testing, integer factorization, the distribution of prime numbers, the Riemann hypothesis, diophantine equations, and the ABC conjecture, but this only begins to scratch the surface. Bach and Shallit's book https://mitpress.mit.edu/books/algorithmic-number-theory-volume-1 goes a bit further, and Henri Cohen has written a slew of books on computational number theory (see the five Springer GTM books listed in his wikipedia entry). To get a sense of the current state of the field, see the discrete mathematics section of Mathematics of Computation (the computational number theory papers generally make up the last 1/3 of each volume) and the proceedings of the biannual Algorithmic Number Theory Symposium, which focuses almost entirely on computational number theory.

I don't have time right now, but at some point I will try to write a proper wikipedia article on this topic (unless someone else beats me to it -- please do). In the short term I may just replace this stub with a new stub based on Pomerance's article, the current stub is simply unacceptable.

Frobitz (talk) 12:40, 1 March 2020 (UTC)

I rewrote the stub, removed the cleanup and disputed tags, and expanded the list of references and additional reading. The stub is now very brief (two sentences), but it is at least grammatically correct, accurate, and sourced, none of which were true of the previous version. I will try to come back and expand it when I have more time.

Frobitz (talk) 14:01, 1 March 2020 (UTC)