Re: Complex a*x^n+b*y^n puzzle
- --- In firstname.lastname@example.org, "djbroadhurst" <d.broadhurst@...> wrote:
>Nice suggestion, David.
> --- In email@example.com,
> "mikeoakes2" <mikeoakes2@> wrote:
> > I do think x or y should be allowed to be negative - why not?
> I think they should be allowed to be complex - why not?
> Complex a*x^n+b*y^n puzzle: Find 4 Gaussian integers a,b,x,y
> such that x and y are not units and a*x^n+b*y^n is a Gaussian
> prime, for 1 <= n <= n_max, with n_max as large as possible.
Must get coding...
(Of course the train of thought then goes:-
or Eisenstein integers,
or algebraic integers of the ring k(sqrt(m)),
- --- In firstname.lastname@example.org,
"Robdine" <robdine@...> wrote:
> any rational prime that can be represented by the sumThere is only rational prime of the form a^2 + b^2 that yields
> of 2 squares (a^2+b^2) will define 4 gaussian primes
precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2.
If a^2 + b^2 is an odd rational prime, we have
8 [sic] asociates of the Gaussian prime z = a + I*b,
since we may mulitply it and its conjugate z = a - I*b
by the 4 units I, -I, -1, 1, obtaining 8 distinct
Gaussian integers that are prime.