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I am looking through the PARI/GP reference card.

Example, listing primes over $\mathbb{Z}$ is standard computer homework exercise [1]:

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How to write a program to find Gaussian prime numbers $\mathfrak{p} = a + bi \in \mathbb{Z}[i]$ with $10^6 < |a|, |b| < 2 \times 10^6$ ?

Related:

hardmath
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cactus314
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  • This quite old post on Gaussian Primes seems closely related. It's focus is on using a sieve type algorithm analogous to Eratosthenes. Some background theory is rehearsed in this post on factoring. – hardmath Jul 14 '21 at 17:42
  • the exercise is to write the program listing the primes. This could also be here https://cs.stackexchange.com/ – cactus314 Jul 14 '21 at 17:57
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    Are you familiar with the characterization of Gaussian primes up to associates? Is the use of Pari/GP (mentioned only in the title) important? – hardmath Jul 14 '21 at 18:28
  • @hardmath Yes so I am specifying a programming language here... – cactus314 Jul 14 '21 at 18:43
  • In Gaussian integers, primes $p \equiv 3 \pmod4$ stay primes and primes $ p \equiv 1 \pmod 4$ can be factored into $p=(a+bi)(a-bi)$ where $a+bi$ and $a-bi$ are Gaussian primes. See this answer for an algorithm to generate those Gaussian prime factors. – arbashn Jul 14 '21 at 19:17
  • If you wish to avoid implementing the logic of primality over the Gaussian integers yourself (see this previous Question Gaussian prime integers, reducing that to primality of rational integers), then you should consult the factor( ) function described under Pari/GP Arithmetic functions. In the range you want, the distinction between proven primes and probable primes is not important. – hardmath Jul 14 '21 at 21:32

1 Answers1

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I will compute below the numbers $\mathfrak p=a+ib$ constrained to the following "triangle" restrictions: $$ 10^6< b<a<2\cdot 10^6\ .$$ (From here build conjugates and associates to get also $\pm a\pm ib$ and $\pm b\pm ia$.)

In particular, $\mathfrak p\not\in\Bbb Z$, so $p:=\mathfrak p\bar{\mathfrak p}=a^2+b^2$ is a prime in $\Bbb Z$ which splits in two parts in the Gaussian integers. Here, $p$ is congruent to $1$ modulo four lying in the interval $J$ between $2\cdot 10^{12}$ and $8\cdot 10^{12}$.

One (slow) algorithm would take all these primes $p$ one by one, write them in the form $p=a^2+b^2$ and check the "triangle" condition. There are roughly some

? floor(intnum(x=2*10^12, 8*10^12, 1/log(x)))
%3 = 205708205954

primes in $J$, and half of them split in $\Bbb Z[i]$. A lot of primes for my taste. Below, there is an old fashioned pari/gp code implementing this slow algorithm. (It avoids the prime $1+i$, just in case somebody wants to relax $b<a$ to $b\le a$...)

The "real job" is done by the pari/gp factorization of Gaussian integers, a prime p in $\Bbb Z$ was slightly changed to I*p to implicitly tell pari/gp to factor over $\Bbb Z[i]$.

{f(N) = local(p, z, a, b, c, count);
count = 0;
forprime(p=2*N^2, 8*N^2,
    if( (p-3) % 4,
        z = factor(I*p)[1, 1];
        a = real(z); b = imag(z);
        if(a < b, c = a; a = b; b = c;);
        if( (N < b) && (a < 2*N), 
            z = a + I*b; count = count + 1;
            print(count, " --> ", z, " with norm ", p);)))}

Then f(100) delivers some $579$ such (normed) primes, and there is a quick end. The last ones are:

570 --> 190 + 189*I with norm 71821
571 --> 194 + 185*I with norm 71861
572 --> 195 + 184*I with norm 71881
573 --> 197 + 182*I with norm 71933
574 --> 195 + 188*I with norm 73369
575 --> 199 + 186*I with norm 74197
576 --> 196 + 191*I with norm 74897
577 --> 196 + 195*I with norm 76441
578 --> 199 + 194*I with norm 77237
579 --> 199 + 196*I with norm 78017
?

Seen from this perspective, this algorithm seems to be "better" than the brute force algorithm considering all $(a,b)$ with $100<b<a<200$, and testing for the primality of $a+ib$. At the cost of having a good primes generator for $\Bbb Z$.

I have also started...

? f(10^6)
1 --> 1000006 + 1000001*I with norm 2000014000037
2 --> 1000008 + 1000003*I with norm 2000022000073
3 --> 1000010 + 1000003*I with norm 2000026000109
4 --> 1000008 + 1000007*I with norm 2000030000113
5 --> 1000013 + 1000002*I with norm 2000030000173
6 --> 1000013 + 1000008*I with norm 2000042000233
7 --> 1000019 + 1000004*I with norm 2000046000377
8 --> 1000019 + 1000014*I with norm 2000066000557
9 --> 1000018 + 1000017*I with norm 2000070000613
10 --> 1000021 + 1000014*I with norm 2000070000637
11 --> 1000026 + 1000009*I with norm 2000070000757
12 --> 1000028 + 1000007*I with norm 2000070000833
13 --> 1000021 + 1000016*I with norm 2000074000697
14 --> 1000027 + 1000010*I with norm 2000074000829

and the primes are slowly printed one by one, as they are fished and tested for relevant...

dan_fulea
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  • I did some homework, I typed factor(29, 1*I) and it returned [ -I 1] and [5 + 2*I 1] and [2 + 5*I 1]. So we learn that $\mathfrak{p} = 2 + 5i$ is a Gaussian prime. – cactus314 Jul 15 '21 at 16:57
  • OK let $\mathfrak{p} = 195 + 176i$ is prime and $\mathfrak{q} = 186 + 185i$ is prime and then $(p) + (q) = (1) = \mathbb{Z}[i]$ as ideals which is to say $\text{GCD}(\mathfrak{p}, \mathfrak{q}) = 1$. Working through the PARI/GP syntax, I have to look up factor(I*p)[1,1] and so forth. – cactus314 Jul 15 '21 at 18:54