cp's OEIS Frontend

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

%I A033212 #88 Feb 17 2022 11:17:06
%S A033212 19,31,61,79,109,139,151,181,199,211,229,241,271,331,349,379,409,421,
%T A033212 439,499,541,571,601,619,631,661,691,709,739,751,769,811,829,859,919,
%U A033212 991,1009,1021,1039,1051,1069,1129,1171,1201,1231,1249,1279,1291,1321,1381
%N A033212 Primes congruent to 1 or 19 (mod 30).
%C A033212 Theorem: Same as primes of the form x^2+15*y^2 (discriminant -60). Proof: Cox, Cor. 2.27, p. 36.
%C A033212 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
%C A033212 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):
%C A033212 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
%C A033212 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
%C A033212 3. The same as primes of the form x^2+5xy-5y^2 (discriminant 45). - _N. J. A. Sloane_, Jun 01 2014
%C A033212 Also primes of the form x^2+7*x*y+y^2 (discriminant 45).
%C A033212 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.
%C A033212 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.
%C A033212 Also, primes which are squares (mod 3*5). Subsequence of A191018. - _David Broadhurst_ and _M. F. Hasler_, Jan 15 2016
%D A033212 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%D A033212 David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
%H A033212 Juan Arias-de-Reyna, <a href="/A033212/b033212.txt">Table of n, a(n) for n = 1..10000</a>
%H A033212 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 A033212 J. B. Tunnell, <a href="/A033212/a033212.txt">Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)</a>
%H A033212 D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%F A033212 a(n) ~ 4n log n. - _Charles R Greathouse IV_, Nov 09 2012
%t A033212 QuadPrimes2[1, 0, 15, 10000] (* see A106856 *)
%t A033212 Select[Prime@Range[250], MemberQ[{1, 19}, Mod[#, 30]] &] (* _Vincenzo Librandi_, Apr 05 2015 *)
%o A033212 (PARI) select(n->n%30==1||n%30==19, primes(100)) \\ _Charles R Greathouse IV_, Nov 09 2012
%o A033212 (PARI) is(p)=issquare(Mod(p,15))&&isprime(p) \\ _M. F. Hasler_, Jan 15 2016
%Y A033212 Cf. A106856, A139643, A106859.
%Y A033212 Primes in A243173 and in A243174.
%Y A033212 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).
%Y A033212 For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K A033212 nonn,easy
%O A033212 1,1
%A A033212 _N. J. A. Sloane_
%E A033212 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.
%E A033212 Typo in crossrefs fixed by _Colin Barker_, Apr 05 2015