

A033212


Primes congruent to 1 or 19 (mod 30).


42



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
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OFFSET

1,1


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^2xy+4y^2 (discriminant 15) and x^2xy+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+5xy5y^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 + (4c1) 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.


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



FORMULA



MATHEMATICA

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


PROG

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


CROSSREFS

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


KEYWORD

nonn,easy


AUTHOR



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.


STATUS

approved



