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

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• ... From: djbroadhurst To: primenumbers@yahoogroups.com Sent: Friday, December 04, 2009 5:31 PM Subject: [PrimeNumbers] Re: Complex a*x^n+b*y^n puzzle thanks
Message 1 of 37 , Dec 5, 2009
----- Original Message -----
Sent: Friday, December 04, 2009 5:31 PM
Subject: [PrimeNumbers] Re: Complex a*x^n+b*y^n puzzle

thanks David,

next conclusion is that any rational prime that can be represented by the sum
of 2 squares (a^2+b^2) will define 4 gaussian primes a+bi,a-bi,b+ai,b-ai.
right?

gr. Rob

> Is a gaussian number a+bi prime if a^2+b^2 is a (integer)prime
> and if c^2+d^2 is prime then is also c+di a gaussian prime?

These two question appear to be identical;
the second merely changes the word order
and the labelling in the first, without
altering the mathematics in any respect.

In each case, the answer is yes.

Moreover, if a and b are non-zero rational integers, then
a + I*b is a Gaussian prime if and *only* if
a^2 + b^2 is a rational prime.

Note, however, that 3 + I*0 is a Gaussian prime,
but 3^2 + 0^2 is not a rational prime.

To determine whether z is a Gaussian prime
we may use the following procedure:

{isgp(z)=local(a,b,c,t=0);
a=abs(real(z));b=abs(imag(z));
if(type(a)=="t_INT"&&type(b)=="t_INT",c=max(a,b);
if(a*b,t=isprime(a^2+b^2),t=(c%4==3)&&isprime(c)));t;}

for(a=1,7,for(b=0,a,z=a+I*b;if(isgp(z),print1(z", "))));

1 + I, 2 + I, 3, 3 + 2*I, 4 + I, 5 + 2*I, 5 + 4*I, 6 + I, 6 + 5*I, 7, 7 + 2*I,

David

[Non-text portions of this message have been removed]
• ... 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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