A141373 - OEIS

%I #57 Sep 08 2022 08:45:35

%S 3,19,43,67,139,163,211,283,307,331,379,499,523,547,571,619,643,691,

%T 739,787,811,859,883,907,1051,1123,1171,1291,1459,1483,1531,1579,1627,

%U 1699,1723,1747,1867,1987,2011,2083,2131,2179,2203,2251,2347,2371,2467,2539

%N Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2).

%C The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

%C Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - _T. D. Noe_, May 19 2008

%D Z. I. Borevich and I. R. Shafarevich, Number Theory.

%H Ray Chandler, <a href="/A141373/b141373.txt">Table of n, a(n) for n = 1..10000</a> [First 1000 terms from Vincenzo Librandi]

%H William C. Jagy and Irving Kaplansky, <a href="/A244019/a244019.pdf">Positive definite binary quadratic forms that represent the same primes</a> [Cached copy] See Table II.

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a>: Index to related sequences, programs, references. OEIS wiki, June 2014.

%H D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.

%F Except for 3, the primes are congruent to 19 (mod 24). - _T. D. Noe_, May 02 2008

%e 19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

%t QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)

%o (Magma) [3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // _Vincenzo Librandi_, Jul 24 2012

%o (PARI) list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\16), if(isprime(t=w+16*y^2), listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, Feb 09 2017

%Y Cf. A107132, A139827, A107003, A141375, A141376 (d=96).

%Y See also A038872 (d=5),

%Y A038873 (d=8),

%Y A068228, A141123 (d=12),

%Y A038883 (d=13),

%Y A038889 (d=17),

%Y A141158 (d=20),

%Y A141159, A141160 (d=21),

%Y A141170, A141171 (d=24),

%Y A141172, A141173 (d=28),

%Y A141174, A141175 (d=32),

%Y A141176, A141177 (d=33),

%Y A141178 (d=37),

%Y A141179, A141180 (d=40),

%Y A141181 (d=41),

%Y A141182, A141183 (d=44),

%Y A033212, A141785 (d=45),

%Y A068228, A141187 (d=48),

%Y A141188 (d=52),

%Y A141189 (d=53),

%Y A141190, A141191 (d=56),

%Y A141192, A141193 (d=57),

%Y A107152, A141302, A141303, A141304 (d=60),

%Y A141215 (d=61),

%Y A141111, A141112 (d=65),

%Y A141336, A141337 (d=92),

%Y A141338, A141339 (d=93),

%Y A141161, A141163 (d=148),

%Y A141165, A141166 (d=229),

%Y A141167, A141167, A141167 (d=257).

%K nonn

%O 1,1

%A _T. D. Noe_, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

%E More terms from _Colin Barker_, Apr 05 2015

%E Edited by _N. J. A. Sloane_, Jul 14 2019, combining two identical entries both with multiple cross-references.