On 23/05/2016 04:39, Keith F. Lynch wrote:
Are those the only two interesting ways of generalizing integers to the complex plane?
No. Let D be any square-free positive integer > 1 and consider numbers of the form p+q.sqrt(d) where p,q are rational; ... You don't mention i. Are these numbers all real?
Whoops. I meant p + q.sqrt(-d). (Or else: let D be any square-free integer other than 1 and, etc. The case where the thing you sqrt() is positive works perfectly well, but indeed it has nothing much to do with complex numbers.)
I once played with numbers in the form a + b*sqrt(2), where a and b are integers. Those appear to form a field.
If you let a,b be rational then they form a field. If you let a,b be integers then they form a ring, which is in fact the ring of algebraic integers contained in that field.
My intention was to find a way to efficiently factor large integers. My plan was to find "second-order primes" which factor ordinary primes,
If d is a quadratic residue mod p (but not a multiple of p) then in (the ring of integers of) the field Q(sqrt(d)) p splits into two primes whose product is p. But you need to be careful in this area -- e.g., in most of these rings you don't have unique factorization, and for some purposes you need to consider not prime *numbers* but things called prime *ideals* which are more complicated. I don't think any of this will help with factoring large numbers, though. (But see e.g. https://en.wikipedia.org/wiki/Quadratic_sieve and https://en.wikipedia.org/wiki/General_number_field_sieve for some powerful factoring techniques that are slightly related...)
Gaussian primes looked like a promising set of second-order primes, except for two things: Only about half of all regular primes have Gaussian prime divisors, and there's no obvious candidate for third-order primes.
Using any quadratic field it turns out that about half of all primes "split". You could get your higher-order primes by looking at larger extensions -- e.g., go from Q(sqrt(2)) to Q(sqrt(2),sqrt(3)). But only half the primes that split the first time will split again the second time, and the computations will get more painful, and I don't think this will help you factor large numbers. (But, important note, I'm not a real number theorist and I could be wrong.) -- g