A191060 -id:A191060 - cp's OEIS frontend

A296920 Rational primes that decompose in the quadratic field Q(sqrt(-11)).

3, 5, 23, 31, 37, 47, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 179, 181, 191, 199, 223, 229, 251, 257, 269, 311, 313, 317, 331, 353, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521, 577, 587, 599, 617, 619, 631, 641, 643, 647, 653, 661, 683, 691, 709, 719

Offset: 1

N. J. A. Sloane, Dec 25 2017

Primes that are 1, 3, 5, 9, or 15 mod 22. - Charles R Greathouse IV, Mar 18 2018

(Which means: union of A141849, A141850, A141852, A141856 and A141851. - R. J. Mathar, Apr 15 2024)

Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.

Cf. A191060, A056874.

# In the quadratic field Q(sqrt(D)), for squarefree D<0, compute lists of:
# rational primes that decompose (SD),
# rational primes that are inert (SI),
# primes p such that D is a square mod p (QR), and
# primes p such that D is a nonsquare mod p (NR),
# omitting the latter if it is the same as the inert primes.
# Consider first M primes p.
# Reference: Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.
with(numtheory):
HH := proc(D,M)
    local SD,SI,QR,NR,p,q,i,t1;
    # if D >= 0 then error("D must be negative"); fi;
    if not issqrfree(D) then
        error("D must be squarefree");
    end if;
    q:=-D;
    SD:=[]; SI:=[]; QR:=[]; NR:=[];
    if (D mod 8) = 1 then
        SD:=[op(SD),2];
    end if;
    if (D mod 8) = 5 then
        SI:=[op(SI),2];
    end if;
    for i from 2 to M do
        p:=ithprime(i);
        if (D mod p) <> 0 and legendre(D,p)=1 then
            SD:=[op(SD),p];
        end if;
        if (D mod p) <> 0 and legendre(D,p)=-1 then
            SI:=[op(SI),p];
        end if;
    end do;
    for i from 1 to M do
        p:=ithprime(i);
        if legendre(D,p) >= 0 then
            QR:=[op(QR),p];
        else
        NR:=[op(NR),p];
        end if;
    end do:
    lprint("Primes that decompose:", SD);
    lprint("Inert primes:", SI);
    lprint("Primes p such that Legendre(D,p) = 0 or 1: ", QR);
    if SI <> NR then
        lprint("Note: SI <> NR here!");
        lprint("Primes p such that Legendre(D,p) = -1: ", NR);
    end if;
end proc:
HH(-11,200); # produces the present sequence (A296920), A191060, and A056874.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[KroneckerSymbol[-11, p] == 1, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Apr 29 2019 *)

list(lim)=my(v=List()); forprime(p=2,lim, if(kronecker(-11,p)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018

a(n) ~ 2n log n. - Charles R Greathouse IV, Mar 18 2018

A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

Original entry on oeis.org

3, 4, 5, 11, 23, 31, 37, 47, 49, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 169, 179, 181, 191, 199, 223, 229, 251, 257, 269, 289, 311, 313, 317, 331, 353, 361, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-11))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 0, 1, 3, 4, 5, 9 modulo 11 and the squares of primes congruent to 2, 6, 7, 8, 10 modulo 5.
For primes p == 1, 3, 4, 5, 9 (mod 11), there are two distinct ideals with norm p in Z[(1+sqrt(-11))/2], namely (x + y*(1+sqrt(-11))/2) and (x + y*(1-sqrt(-11))/2), where (x,y) is a solution to x^2 + x*y + 3*y^2 = p; for p = 11, (sqrt(-11)) is the unique ideal with norm p; for p == 2, 6, 7, 8, 10 (mod 11), (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-11))/2) = norm((1 - sqrt(-11))/2) = 3;
norm((3 + sqrt(-11))/2) = norm((3 - sqrt(-11))/2) = 5;
norm((9 + sqrt(-11))/2) = norm((9 - sqrt(-11))/2) = 23;
norm((5 + 3*sqrt(-11))/2) = norm((5 - 3*sqrt(-11))/2) = 31.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011582, A056874, A296920, A191060.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035179.
The total number of elements with norm n is given by A028609.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), this sequence (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341785(n) = my(disc=-11); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A369863 Inert rational primes in the field Q(sqrt(-21)).

Original entry on oeis.org

13, 29, 43, 47, 53, 59, 61, 67, 73, 79, 83, 97, 113, 127, 131, 137, 149, 151, 157, 163, 167, 181, 197, 211, 227, 229, 233, 241, 251, 281, 311, 313, 317, 331, 349, 379, 383, 389, 397, 401, 409, 419, 433, 449, 463, 467, 479, 487, 499, 503, 547, 557, 563, 569, 571, 577, 587

Offset: 1

Dimitris Cardaris, Feb 03 2024

nonn

Primes p such that Legendre(-21,p) = -1.

Cf. inert rational primes in the imaginary quadratic field Q(sqrt(-d)) for the first squarefree positive integers d: A002145 (1), A003628 (2), A003627 (3), A003626 (5), A191059 (6), A003625 (7), A296925 (10), A191060 (11), A105885 (13), A191061 (14), A191062 (15), A296930 (17), A191063 (19), this sequence (21), A191064 (22), A191065 (23).

Select[Range[3,600], PrimeQ[#] && JacobiSymbol[-21,#]==-1 &] (* Stefano Spezia, Feb 04 2024 *)

[p for p in prime_range(3, 600) if legendre_symbol(-21, p) == -1]

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A296920 Rational primes that decompose in the quadratic field Q(sqrt(-11)).

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A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

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A369863 Inert rational primes in the field Q(sqrt(-21)).

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