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

Author

N. J. A. Sloane, Dec 25 2017

Keywords

• nonn
• easy

Comments

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)

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

Cf. A191060, A056874.

Programs

Maple

# 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.

Mathematica

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 *)

PARI

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

Formula

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