cp's OEIS Frontend

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

%I A296920 #27 Apr 15 2024 05:31:43
%S A296920 3,5,23,31,37,47,53,59,67,71,89,97,103,113,137,157,163,179,181,191,
%T A296920 199,223,229,251,257,269,311,313,317,331,353,367,379,383,389,397,401,
%U A296920 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 A296920 Rational primes that decompose in the quadratic field Q(sqrt(-11)).
%C A296920 Primes that are 1, 3, 5, 9, or 15 mod 22. - _Charles R Greathouse IV_, Mar 18 2018
%C A296920 (Which means: union of A141849, A141850, A141852, A141856 and A141851. - _R. J. Mathar_, Apr 15 2024)
%D A296920 Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.
%H A296920 Robert Israel, <a href="/A296920/b296920.txt">Table of n, a(n) for n = 1..10000</a>
%H A296920 <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 A296920 a(n) ~ 2n log n. - _Charles R Greathouse IV_, Mar 18 2018
%p A296920 # In the quadratic field Q(sqrt(D)), for squarefree D<0, compute lists of:
%p A296920 # rational primes that decompose (SD),
%p A296920 # rational primes that are inert (SI),
%p A296920 # primes p such that D is a square mod p (QR), and
%p A296920 # primes p such that D is a nonsquare mod p (NR),
%p A296920 # omitting the latter if it is the same as the inert primes.
%p A296920 # Consider first M primes p.
%p A296920 # Reference: Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.
%p A296920 with(numtheory):
%p A296920 HH := proc(D,M)
%p A296920     local SD,SI,QR,NR,p,q,i,t1;
%p A296920     # if D >= 0 then error("D must be negative"); fi;
%p A296920     if not issqrfree(D) then
%p A296920         error("D must be squarefree");
%p A296920     end if;
%p A296920     q:=-D;
%p A296920     SD:=[]; SI:=[]; QR:=[]; NR:=[];
%p A296920     if (D mod 8) = 1 then
%p A296920         SD:=[op(SD),2];
%p A296920     end if;
%p A296920     if (D mod 8) = 5 then
%p A296920         SI:=[op(SI),2];
%p A296920     end if;
%p A296920     for i from 2 to M do
%p A296920         p:=ithprime(i);
%p A296920         if (D mod p) <> 0 and legendre(D,p)=1 then
%p A296920             SD:=[op(SD),p];
%p A296920         end if;
%p A296920         if (D mod p) <> 0 and legendre(D,p)=-1 then
%p A296920             SI:=[op(SI),p];
%p A296920         end if;
%p A296920     end do;
%p A296920     for i from 1 to M do
%p A296920         p:=ithprime(i);
%p A296920         if legendre(D,p) >= 0 then
%p A296920             QR:=[op(QR),p];
%p A296920         else
%p A296920         NR:=[op(NR),p];
%p A296920         end if;
%p A296920     end do:
%p A296920     lprint("Primes that decompose:", SD);
%p A296920     lprint("Inert primes:", SI);
%p A296920     lprint("Primes p such that Legendre(D,p) = 0 or 1: ", QR);
%p A296920     if SI <> NR then
%p A296920         lprint("Note: SI <> NR here!");
%p A296920         lprint("Primes p such that Legendre(D,p) = -1: ", NR);
%p A296920     end if;
%p A296920 end proc:
%p A296920 HH(-11,200); # produces the present sequence (A296920), A191060, and A056874.
%t A296920 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 A296920 (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 A296920 Cf. A191060, A056874.
%K A296920 nonn,easy
%O A296920 1,1
%A A296920 _N. J. A. Sloane_, Dec 25 2017