- --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>

Nice suggestion, David.

> --- In primenumbers@yahoogroups.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)),

or... :-)

Mike - --- In primenumbers@yahoogroups.com,

"Robdine" <robdine@...> wrote:

> any rational prime that can be represented by the sum

There 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.

David