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A296920
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Rational primes that decompose in the quadratic field Q(sqrt(-11)).
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29
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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
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OFFSET
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1,1
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COMMENTS
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REFERENCES
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Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.
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LINKS
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FORMULA
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MAPLE
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# 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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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