A068228 Primes congruent to 1 (mod 12).

13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 937, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297

Offset: 1

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Keywords

• easy
• nonn

Comments

This has several equivalent definitions (cf. the Tunnell link)

Also primes of the form x^2 + 9y^2 (discriminant -36). - T. D. Noe, May 07 2005 [corrected by Klaus Purath, Jan 18 2023]

Also primes of the form x^2 - 12y^2 (discriminant 48). Cf. A140633. - T. D. Noe, May 19 2008 [corrected by Klaus Purath, Jan 18 2023]

Also primes of the form x^2 + 4*x*y + y^2.

Also primes of the form x^2 + 2*x*y - 2*y^2 (cf. A084916).

Also primes of the form x^2 + 6*x*y - 3*y^2.

Also primes of the form 4*x^2 + 8*x*y + y^2.

Also primes of the form u^2 - 3v^2 (use the transformation {u,v} = {x+2y,y}). - Tito Piezas III, Dec 28 2008

Sequence lists generalized cuban primes (A007645) that are the sum of 2 nonzero squares. - Altug Alkan, Nov 25 2015

Yasutoshi Kohmoto observes that prevprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the previous prime must be at a gap of 4 or 8 or 12 ..., but a gap of 4 is impossible because 12k + 1 - 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the previous prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs. 65% as the above simple explanation suggests, but considering primes up to 10^8 yields a ratio of about 41% vs. 59%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017

Also primes of the form x^2 - 27*y^2. - Klaus Purath, Jan 18 2023

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy, with permission]
• Michael Penn, an example right from my number theory class., YouTube video, 2021.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
• J. Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

Cf. A068227, A068229, A040117, A068231, A068232, A068233, A068234, A068235, A139643, A141122, A140633, A264732.

Subsequence of A084916.

Subsequence of A007645.

Also primes in A084916, A020672.

Cf. A141123 (d=12), A141111, A141112 (d=65), A141187 (d=48) A038872 (d=5), A038873 (d=8), A038883 (d=13), A038889 (d=17).

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

Programs

Magma

[p: p in PrimesUpTo(1400) | p mod 12 in {1}]; // Vincenzo Librandi, Jul 14 2012
For other programs see the "Binary Quadratic Forms and OEIS" link.

Maple

select(isprime, [seq(i,i=1..10000, 12)]); # Robert Israel, Nov 27 2015

Mathematica

Select[Prime/@Range[250], Mod[ #, 12]==1&]
Select[Range[13, 10^4, 12], PrimeQ] (* Zak Seidov, Mar 21 2011 *)

PARI

for(i=1,250, if(prime(i)%12==1, print(prime(i))))

PARI

forstep(p=13,10^4,12,isprime(p)&print(p)); \\ Zak Seidov, Mar 21 2011

Extensions

Edited by Dean Hickerson, Feb 27 2002

Entry revised by N. J. A. Sloane, Oct 18 2014 (Edited, merged with A141122, 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 05 2008).