A141373 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).

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

Offset: 1

Author

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

Keywords

• nonn

Comments

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.

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

Examples

			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).
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
• William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy] See Table II.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

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

See also A038872 (d=5),

A038873 (d=8),

A068228, A141123 (d=12),

A038883 (d=13),

A038889 (d=17),

A141158 (d=20),

A141159, A141160 (d=21),

A141170, A141171 (d=24),

A141172, A141173 (d=28),

A141174, A141175 (d=32),

A141176, A141177 (d=33),

A141178 (d=37),

A141179, A141180 (d=40),

A141181 (d=41),

A141182, A141183 (d=44),

A033212, A141785 (d=45),

A068228, A141187 (d=48),

A141188 (d=52),

A141189 (d=53),

A141190, A141191 (d=56),

A141192, A141193 (d=57),

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

A141215 (d=61),

A141111, A141112 (d=65),

A141336, A141337 (d=92),

A141338, A141339 (d=93),

A141161, A141163 (d=148),

A141165, A141166 (d=229),

A141167, A141167, A141167 (d=257).

Programs

Magma

[3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

Mathematica

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

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

Formula

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

Extensions

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.