## Re: Complex a*x^n+b*y^n puzzle

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• ... Nice suggestion, David. Must get coding... (Of course the train of thought then goes:- or Eisenstein integers, or algebraic integers of the ring
Message 1 of 37 , Nov 28 5:06 AM
>
> "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.

Nice suggestion, David.
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
• ... There is only rational prime of the form a^2 + b^2 that yields precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2. If a^2 + b^2 is an odd rational
Message 37 of 37 , Dec 5, 2009
"Robdine" <robdine@...> wrote:

> any rational prime that can be represented by the sum
> of 2 squares (a^2+b^2) will define 4 gaussian primes

There is only rational prime of the form a^2 + b^2 that yields
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
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