Complex Prime Number Test - Printable Version

Complex Prime Number Test - Printable Version

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Complex Prime Number Test - Eddie W. Shore - 11-11-2014 06:09 PM

Program:

Code:

EXPORT cisprime(z)

BEGIN

// 2014-11-09 EWS

LOCAL a,b,r,t;

a:=RE(z);

b:=IM(z);

r:=a^2+b^2;

IF CAS.isPrime(r)==1 THEN

t:=1; END;

IF a==0 AND ABS(b) MOD 4==3 AND CAS.isprime(ABS(b))==1 THEN

t:=1; END;

IF b==0 AND ABS(a) MOD 4==3 AND CAS.isprime(ABS(a))==1 THEN

t:=1; END;

RETURN t;

END;

(Edited 2014-11-13, see the thread - thanks Joe and Thomas!)

Examples:
cisprime(2-4*i) returns 0
cisprime(-2+5*i) returns 1

Source: http://mathworld.wolfram.com/GaussianPrime.html

RE: Complex Prime Number Test - Jim Horn - 11-12-2014 04:41 PM

Wow - never heard of those before. Thanks for teaching us!

I am puzzled - your program tests for 3mod4 of the non-zero component if the other is zero, but doesn't test the non-zero component for primality. So, for example, 0+/- 15i isn't a Gaussian prime as 15 isn't prime. Am I mistaken?

RE: Complex Prime Number Test - Thomas Klemm - 11-12-2014 05:31 PM

(11-12-2014 04:41 PM)Jim Horn Wrote: I am puzzled - your program tests for 3mod4 of the non-zero component if the other is zero, but doesn't test the non-zero component for primality. So, for example, 0+/- 15i isn't a Gaussian prime as 15 isn't prime. Am I mistaken?

Quote:If a=0, then bi is a Gaussian prime iff |b| is an ordinary prime and |b|=3 (mod 4).

Thus 15i is not a Gaussian prime since 15 is not an ordinary prime.
However 5 is not a Gaussian prime though it is an ordinary prime since |5|=1 (mod 4).
The complex factors of 5 are (2 + i)(2 − i).

HTH
Thomas

RE: Complex Prime Number Test - Eddie W. Shore - 11-13-2014 01:23 PM

(11-12-2014 05:31 PM)Thomas Klemm Wrote:
(11-12-2014 04:41 PM)Jim Horn Wrote: I am puzzled - your program tests for 3mod4 of the non-zero component if the other is zero, but doesn't test the non-zero component for primality. So, for example, 0+/- 15i isn't a Gaussian prime as 15 isn't prime. Am I mistaken?

Quote:If a=0, then bi is a Gaussian prime iff |b| is an ordinary prime and |b|=3 (mod 4).

Thus 15i is not a Gaussian prime since 15 is not an ordinary prime.
However 5 is not a Gaussian prime though it is an ordinary prime since |5|=1 (mod 4).
The complex factors of 5 are (2 + i)(2 − i).

HTH
Thomas

Thomas,

Thanks for pointing out the glaring omission - much appreciated. The edited algorithm should be correct now.

Eddie