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A296920 Rational primes that decompose in the quadratic field Q(sqrt(-11)). 29
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

FORMULA

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

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"); fi;

q:=-D;

SD:=[]; SI:=[]; QR:=[]; NR:=[];

if (D mod 8) = 1 then SD:=[op(SD), 2]; fi;

if (D mod 8) = 5 then SI:=[op(SI), 2]; fi;

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]; fi;

if (D mod p) <> 0 and legendre(D, p)=-1 then SI:=[op(SI), p]; fi;

od;

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]; fi; od:

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);

fi;

end;

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-Fran├žois Alcover, Apr 29 2019 *)

PROG

(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

CROSSREFS

Cf. A191060, A056874.

Sequence in context: A222424 A067256 A136891 * A106857 A106307 A106282

Adjacent sequences:  A296917 A296918 A296919 * A296921 A296922 A296923

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 25 2017

STATUS

approved

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Last modified May 23 23:52 EDT 2022. Contains 353993 sequences. (Running on oeis4.)