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%I #89 Oct 03 2023 10:07:54
%S 13,37,61,73,97,109,157,181,193,229,241,277,313,337,349,373,397,409,
%T 421,433,457,541,577,601,613,661,673,709,733,757,769,829,853,877,937,
%U 997,1009,1021,1033,1069,1093,1117,1129,1153,1201,1213,1237,1249,1297
%N Primes congruent to 1 (mod 12).
%C This has several equivalent definitions (cf. the Tunnell link)
%C Also primes of the form x^2 + 9y^2 (discriminant -36). - _T. D. Noe_, May 07 2005 [corrected by _Klaus Purath_, Jan 18 2023]
%C 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]
%C Also primes of the form x^2 + 4*x*y + y^2.
%C Also primes of the form x^2 + 2*x*y - 2*y^2 (cf. A084916).
%C Also primes of the form x^2 + 6*x*y - 3*y^2.
%C Also primes of the form 4*x^2 + 8*x*y + y^2.
%C Also primes of the form u^2 - 3v^2 (use the transformation {u,v} = {x+2y,y}). - _Tito Piezas III_, Dec 28 2008
%C Sequence lists generalized cuban primes (A007645) that are the sum of 2 nonzero squares. - _Altug Alkan_, Nov 25 2015
%C 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
%C Also primes of the form x^2 - 27*y^2. - _Klaus Purath_, Jan 18 2023
%D Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%D David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
%H Vincenzo Librandi, <a href="/A068228/b068228.txt">Table of n, a(n) for n = 1..1000</a>
%H William C. Jagy and Irving Kaplansky, <a href="/A244019/a244019.pdf">Positive definite binary quadratic forms that represent the same primes</a> [Cached copy, with permission]
%H Michael Penn, <a href="https://www.youtube.com/watch?v=xVhuazrwYLI">an example right from my number theory class.</a>, YouTube video, 2021.
%H 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)
%H 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 J. Voight, <a href="http://dx.doi.org/10.1090/S0025-5718-07-01976-X">Quadratic forms that represent almost the same primes</a>, Math. Comp., Vol. 76 (2007), pp. 1589-1617.
%H D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%p select(isprime, [seq(i,i=1..10000, 12)]); # _Robert Israel_, Nov 27 2015
%t Select[Prime/@Range[250], Mod[ #, 12]==1&]
%t Select[Range[13, 10^4, 12], PrimeQ] (* _Zak Seidov_, Mar 21 2011 *)
%o (PARI) for(i=1,250, if(prime(i)%12==1, print(prime(i))))
%o (PARI) forstep(p=13,10^4,12,isprime(p)&print(p)); \\ _Zak Seidov_, Mar 21 2011
%o (Magma) [p: p in PrimesUpTo(1400) | p mod 12 in {1}]; // _Vincenzo Librandi_, Jul 14 2012
%o For other programs see the "Binary Quadratic Forms and OEIS" link.
%Y Cf. A068227, A068229, A040117, A068231, A068232, A068233, A068234, A068235, A139643, A141122, A140633, A264732.
%Y Subsequence of A084916.
%Y Subsequence of A007645.
%Y Also primes in A084916, A020672.
%Y Cf. A141123 (d=12), A141111, A141112 (d=65), A141187 (d=48) A038872 (d=5), A038873 (d=8), A038883 (d=13), A038889 (d=17).
%Y For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K easy,nonn
%O 1,1
%A Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002
%E Edited by _Dean Hickerson_, Feb 27 2002
%E 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).
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