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

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• ... Here is the output from complexification of yesterday s simple-minded pari-gp script, for n_max=9:- a=3 + 3*I, b=2 + 3*I, x=2, y=1 - 2*I n=1 u=14 + 5*I
Message 1 of 37 , Nov 28, 2009
>
> 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.

Here is the output from complexification of yesterday's simple-minded pari-gp script, for n_max=9:-

a=3 + 3*I, b=2 + 3*I, x=2, y=1 - 2*I
n=1 u=14 + 5*I uu=221
n=2 u=18 - 5*I uu=349
n=3 u=-4 - 5*I uu=41
n=4 u=-38 + 75*I uu=7069
n=5 u=64 + 295*I uu=91121
n=6 u=558 + 455*I uu=518389
n=7 u=1276 - 85*I uu=1635401
n=8 u=722 - 1485*I uu=2726509
n=9 u=-3016 - 625*I uu=9486881

(u is your expression a*x^n+b*y^n, uu is its norm.)

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 7:49 AM
"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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