Re: Pythagorean Primitives

On 25 Jun 2025 19:05:44 +0100 (BST), Gareth Taylor wrote:

Warning, incoming maths!  But I've tried to make it friendly and
legible.
 
It comes down to factorisations in the "Gaussian integers", which are
numbers of the form a+bi, with a, b integers and i = sqrt(-1).
 
For example, using these we can now factorise 5 = (2+i)(2-i), but we
can't factorise 7 any more than it already is.
 
It can be shown (proof omitted!) that all integer primes which are 1 mod
4 factorise as (a+bi)(a-bi) for some a, b, and moreover that there is
only one such factorisation.  (Well, only one up to moving irrelevant
factors of -1 or +-i around, e.g. writing 5 = (1+2i)(1-2i) instead). The
prime 2 also factorises, as (1+i)(1-i), but that's not going to matter
to us.

Thanks Gareth for the comprehensive and clear explanation.

That's an great insight.



--
David Entwistle