A033212 Primes congruent to 1 or 19 (mod 30).

19, 31, 61, 79, 109, 139, 151, 181, 199, 211, 229, 241, 271, 331, 349, 379, 409, 421, 439, 499, 541, 571, 601, 619, 631, 661, 691, 709, 739, 751, 769, 811, 829, 859, 919, 991, 1009, 1021, 1039, 1051, 1069, 1129, 1171, 1201, 1231, 1249, 1279, 1291, 1321, 1381

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Theorem: Same as primes of the form x^2+15*y^2 (discriminant -60). Proof: Cox, Cor. 2.27, p. 36.

Equivalently, primes congruent to 1 or 4 (mod 15). Also x^2+xy+4y^2 is the principal form of (fundamental) discriminant -15. The only other class for -15 contains the form 2x^2+xy+2y^2 (A106859), in the other genus. - Rick L. Shepherd, Jul 25 2014

Three further theorems (these were originally stated as conjectures, but are now known to be theorems, thanks to the work of J. B. Tunnell - see link):

1. The same as primes of the form x^2-xy+4y^2 (discriminant -15) and x^2-xy+19y^2 (discriminant -75), both with x and y nonnegative. - T. D. Noe, Apr 29 2008

2. The same as primes of the form x^2+xy+19y^2 (discriminant -75), with x and y nonnegative. - T. D. Noe, Apr 29 2008

3. The same as primes of the form x^2+5xy-5y^2 (discriminant 45). - N. J. A. Sloane, Jun 01 2014

Also primes of the form x^2+7*x*y+y^2 (discriminant 45).

Lemma (Will Jagy, Jun 12 2014): If c is any (positive or negative) even number, then x^2 + x y + c y^2 and x^2 + (4 c - 1) y^2 represent the same odd numbers.

Proof: x (x + y) + c y^2 = odd, therefore x is odd, x + y odd, so y is even. Let y = 2 t. Then x( x + 2 t) + 4 c t^2 = x^2 + 2 x t + 4 c t^2 = (x+t)^2 + (4c-1) t^2 = odd. QED With c = 4, neither one represents 2, so x^2+15y^2 and x^2+xy+4y^2 represent the same primes.

Also, primes which are squares (mod 3*5). Subsequence of A191018. - David Broadhurst and M. F. Hasler, Jan 15 2016

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
• David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
• J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

Cf. A106856, A139643, A106859.

Primes in A243173 and in A243174.

Cf. A141785 (d=45), A033212 (Primes of form x^2+15*y^2), A038872(d=5), A038873 (d=8), A068228, A141123 (d=12), A038883 (d=13), A038889 (d=17), A141111, A141112 (d=65).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

Mathematica

QuadPrimes2[1, 0, 15, 10000] (* see A106856 *)
Select[Prime@Range[250], MemberQ[{1, 19}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)

PARI

select(n->n%30==1||n%30==19, primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

PARI

is(p)=issquare(Mod(p,15))&&isprime(p) \\ M. F. Hasler, Jan 15 2016

Formula

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

Extensions

Edited by N. J. A. Sloane, Jun 01 2014 and Oct 18 2014: added Tunnell document, revised entry, merged with A141184. The latter entry was submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008.

Typo in crossrefs fixed by Colin Barker, Apr 05 2015