Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> 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; this forms a field; the algebraic integers in the field are either the numbers of the form p+q.sqrt(d) where p,q are integers, or the numbers of the form p+q(1+sqrt(d))/2 where p,q are integers, and these are by any reasonable criteria an "interesting way of generalizing integers to the complex plane".
You don't mention i. Are these numbers all real? I once played with numbers in the form a + b*sqrt(2), where a and b are integers. Those appear to form a 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, "third-order primes" which factor second-order primes, etc. Then I'd find which of the first thousand or so, say, tenth-order primes divide the number I'm trying to factor. I'd then combine them by pairs or triples to find ninth-order primes which would of course also factor the target number. I'd then combine those by pairs or triples to find eighth-order primes which would of course also factor the target number. And I'd work my way up until I had the first-order (regular integer) primes which factor the target number. I'm indifferent to the nature of these higher-order primes. They could be complex, quaternions, algebraic, matrices, or something entirely new and weird. And the operation might not even be ordinary multiplication except in the last step. 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. One of my approaches was to form a field of two-part real algebraic numbers in the form a + b*sqrt(2) for second-order primes, another field consisting of four-part numbers based on the 4th roots of 2 for third-order primes, another field consisting of eight-part numbers based on the 8th roots of 2 for fourth-order primes, etc. But it turned out that the only regular prime any of these numbers would non-trivially divide was 2 itself. Sticking in cube roots of 2, square roots of 3, etc., didn't help much.