A319447 a(n) is the rank of the multiplicative group of Eisenstein integers modulo n.

0, 1, 1, 2, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 4, 2, 3, 3, 2, 1, 4, 2, 3, 3, 4, 1, 3, 2, 3, 2, 2, 3, 4, 2, 3, 3, 4, 1, 4, 2, 3, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 5, 3, 2, 1, 4, 2, 3, 5, 3, 3, 3, 2, 3, 2, 4, 1, 4, 2, 3, 2, 4, 3, 4, 2, 4, 3, 2, 1, 5, 2, 3, 2

Offset: 1

Author

Jianing Song, Sep 19 2018

Keywords

• nonn

Comments

The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G.

Let p be an odd prime and (Z[w]/nZ[w])* be the multiplicative group of Gaussian integers modulo n, then: (Z[w]/p^e*Z[w])* = (C_((p-1)*p^(e-1)))^2 if p == 1 (mod 6); C_(p^(e-1)) X C_(p^(e-1)*(p^2-1)) if p == 5 (mod 6); (Z[w]/3^e*Z[w])* = C_3 X C_(3^(e-1)) X C_(2*3^(e-1)); (Z[w]/2Z[w])* = C_3, (Z[w]/2^e*Z[w])* = C_2 X C_(2^(e-2)) X C_(3*2^(e-1)) for e >= 2. If n = Product_{i=1..k} (p_i)^(e_i), then (Z[w]/nZ[w])* = (Z[w]/(p_1)^(e_1)*Z[w])* X (Z[w]/(p_2)^(e_2)*Z[w])* X ... X (Z[w]/(p_k)^(e_k)*Z[w])*.

The order of (Z[w]/nZ[w])* is A319445(n) and the exponent of it is A319446(n).

{a(n)} is not additive: (Z[w]/2Z[w])* = C_3, (Z[w]/25Z[w])* = C_5 X C_120, so (Z[w]/50Z[w])* = C_15 X C_120, a(50) < a(2) + a(25).

A319445(n)/A319446(n) is always an integer, and is 1 if and only if (Z[w]/nZ[w])* is cyclic, that is, rank((Z[w]/nZ[w])*) = a(n) = 0 or 1, and n has a primitive root in (Z[w]/nZ[w])*. a(n) = 1 if and only if n = 3 or a prime congruent to 2 mod 3. - Jianing Song, Jan 08 2019

From Jianing Song, Oct 03 2022: (Start)

More generally, let pi be a prime element of Z[w] of norm p or p^2 for prime p, then:

- for p == 1 (mod 6), (Z[w]/(pi^e)Z[w])* = C_((p-1)*p^(e-1));

- for p == 5 (mod 6), (Z[w]/(pi^e)Z[w])* = C_(p^(e-1)) X C_(p^(e-1)*(p^2-1));

- for p = 3, (Z[w]/(pi^e)Z[w])* = C_2 for e = 1, C_3 X C_(3^floor((e-2)/2)) X C_(2*3^ceiling((e-2)/2)) for e >= 2;

- for p = 2, (Z[w]/(pi^e)Z[w])* = C_3 for e = 1, C_2 X C_(2^(e-2)) X C_(3*2^(e-1)) for e >= 2.

For a more general result see my link below. (End)

Examples

			(Z[w]/1Z[w])* = C_1 (has rank 0);
(Z[w]/2Z[w])* = C_3 (has rank 1);
(Z[w]/3Z[w])* = C_6 (has rank 1);
(Z[w]/4Z[w])* = C_2 X C_6 (has rank 2);
(Z[w]/5Z[w])* = C_24 (has rank 1);
(Z[w]/6Z[w])* = C_3 X C_6 (has rank 2);
(Z[w]/7Z[w])* = C_6 X C_6 (has rank 2);
(Z[w]/8Z[w])* = C_2 X C_2 X C_12 (has rank 3);
(Z[w]/9Z[w])* = C_3 X C_3 X C_6 (has rank 3);
(Z[w]/10Z[w])* = C_3 X C_24 (has rank 2).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Jianing Song, Structure of (R/P^e)* for R the ring of integers of a quadratic field and P a prime ideal of R
• Wikipedia, Eisenstein integer

Crossrefs

Cf. A046072, A319445, A319446.

Equivalent in the ring of Gaussian integers: A316506.

Programs

PARI

rad(n) = factorback(factorint(n)[, 1]);
grad(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2&e==1, r*=3);
        if(p==2&e==2, r*=12);
        if(p==2&e>=3, r*=24);
        if(p==3&e==1, r*=6);
        if(p==3&e>=2, r*=54);
        if(p%6==1, r*=(rad(p-1))^2);
        if(p%6==5&e==1, r*=rad(p^2-1));
        if(p%6==5&e>=2, r*=p^2*rad(p^2-1));
    );
    return(r);
}
a(n)=if(n>1, vecmax(factor(grad(n))[, 2]), 0); \\ The program is based on the facts that although rank((Z[w]/nZ[w])*) is not additive, the p-rank of (Z[w]/nZ[w])* is additive for any prime p, and that rank((Z[w]/nZ[w])*) is the maximum of the p-rank of (Z[w]/nZ[w])* where p runs through all primes. - Jianing Song, Aug 05 2019

Formula

Let p be an odd prime, then: a(p^e) = 2 if p == 1 (mod 6) or p == 5 (mod 6), e >= 2; a(p) = 1 if p == 5 (mod 6). a(3) = 1, a(3^e) = 3 for e >= 2. a(2) = 1, a(4) = 2, a(2^e) = 3 for e >= 3. [Corrected by Jianing Song, Aug 05 2019]

Extensions

Corrected by Jianing Song, Jan 12 2019