A296920 - OEIS

%I #27 Apr 15 2024 05:31:43

%S 3,5,23,31,37,47,53,59,67,71,89,97,103,113,137,157,163,179,181,191,

%T 199,223,229,251,257,269,311,313,317,331,353,367,379,383,389,397,401,

%U 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

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

%C Primes that are 1, 3, 5, 9, or 15 mod 22. - _Charles R Greathouse IV_, Mar 18 2018

%C (Which means: union of A141849, A141850, A141852, A141856 and A141851. - _R. J. Mathar_, Apr 15 2024)

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

%H Robert Israel, <a href="/A296920/b296920.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>

%F a(n) ~ 2n log n. - _Charles R Greathouse IV_, Mar 18 2018

%p # In the quadratic field Q(sqrt(D)), for squarefree D<0, compute lists of:

%p # rational primes that decompose (SD),

%p # rational primes that are inert (SI),

%p # primes p such that D is a square mod p (QR), and

%p # primes p such that D is a nonsquare mod p (NR),

%p # omitting the latter if it is the same as the inert primes.

%p # Consider first M primes p.

%p # Reference: Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.

%p with(numtheory):

%p HH := proc(D,M)

%p     local SD,SI,QR,NR,p,q,i,t1;

%p     # if D >= 0 then error("D must be negative"); fi;

%p     if not issqrfree(D) then

%p         error("D must be squarefree");

%p     end if;

%p     q:=-D;

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

%p     if (D mod 8) = 1 then

%p         SD:=[op(SD),2];

%p     end if;

%p     if (D mod 8) = 5 then

%p         SI:=[op(SI),2];

%p     end if;

%p     for i from 2 to M do

%p         p:=ithprime(i);

%p         if (D mod p) <> 0 and legendre(D,p)=1 then

%p             SD:=[op(SD),p];

%p         end if;

%p         if (D mod p) <> 0 and legendre(D,p)=-1 then

%p             SI:=[op(SI),p];

%p         end if;

%p     end do;

%p     for i from 1 to M do

%p         p:=ithprime(i);

%p         if legendre(D,p) >= 0 then

%p             QR:=[op(QR),p];

%p         else

%p         NR:=[op(NR),p];

%p         end if;

%p     end do:

%p     lprint("Primes that decompose:", SD);

%p     lprint("Inert primes:", SI);

%p     lprint("Primes p such that Legendre(D,p) = 0 or 1: ", QR);

%p     if SI <> NR then

%p         lprint("Note: SI <> NR here!");

%p         lprint("Primes p such that Legendre(D,p) = -1: ", NR);

%p     end if;

%p end proc:

%p HH(-11,200); # produces the present sequence (A296920), A191060, and A056874.

%t 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 *)

%o (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

%Y Cf. A191060, A056874.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Dec 25 2017