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%I #95 Oct 31 2022 11:48:31
%S 5,11,19,29,31,41,59,61,71,79,89,101,109,131,139,149,151,179,181,191,
%T 199,211,229,239,241,251,269,271,281,311,331,349,359,379,389,401,409,
%U 419,421,431,439,449,461,479,491,499,509,521,541,569,571,599,601,619
%N Primes congruent to {0, 1, 4} mod 5.
%C Also odd primes p such that 5 is a square mod p: (5/p) = +1 for p > 5.
%C Primes of the form x^2 + x*y - y^2 (as well as of the form x^2 + 3*x*y + y^2), both with discriminant = 5 and class number = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1. [This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales, Jun 06 2008. _R. J. Mathar_ proved on Jul 22 2008 that this coincides with the present sequence.]
%C Also primes of the form 5x^2 - y^2 (cf. A031363). - _N. J. A. Sloane_, May 30 2014
%C Also primes of the form x^2 + 4*x*y - y^2. Every binary quadratic primitive form of discriminant 20 or 5 has proper solutions for positive integers N given in A089270, including the present primes. Proof from computing the corresponding representative parallel primitive forms, which leads to x^2 - 5 == 0 (mod N) or x^2 + x - 1 == 0 (mod N) which have solutions precisely for these positive N values, including these primes. - _Wolfdieter Lang_, Jun 19 2019
%C For a Pythagorean triple a, b, c, these primes (and 2) are the possible prime factors of 2a + b, |2a - b|, 2b + a, and 2b - a. - _J. Lowell_, Nov 05 2011
%C The prime factors of A028387(n^2+3n+1). - _Richard R. Forberg_, Dec 12 2014
%C a(n) = A045468(n-1) for n > 1. - _Robert Israel_, Dec 22 2014
%C Except for p = 5, these are primes p that divide Fibonacci(p-1). - _Dmitry Kamenetsky_, Jul 27 2015
%C Apart from the first term, these are rational primes that decompose in the field Q[sqrt(5)]. For example, 11 = ((7 + sqrt(5))/2)*((7 - sqrt(5))/2), 19 = ((9 + sqrt(5))/2)*((9 - sqrt(5))/2). - _Jianing Song_, Nov 23 2018
%C The possible prime factors of x^2 - x - 1. - _Charles R Greathouse IV_, Mar 18 2022
%D Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H T. D. Noe, <a href="/A038872/b038872.txt">Table of n, a(n) for n = 1..1000</a>
%H C. Banderier, <a href="http://algo.inria.fr/banderier/Recipro/node19.html">Calcul de (5/p)</a>
%H Henri Darmon, <a href="https://www.ams.org/publications/journals/notices/201703/rnoti-p209.pdf">Andrew Wiles’s Marvelous Proof</a>, Notices of the AMS (2017), Volume 64, Number 3 pp. 209-216. See p. 211.
%H Tamara M. Lavshuk, <a href="https://mzsvfu.ru/index.php/mz/article/view/regular-polygons-and-polyhedra-over-finite-field">Regular polygons and polyhedra over finite field</a>, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.
%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 D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%F a(n) ~ 2n*log(n). - _Charles R Greathouse IV_, Nov 29 2016
%p select(isprime, [5, seq(op([5*k-1,5*k+1]),k=1..1000)]); # _Robert Israel_, Dec 22 2014
%t Join[{5}, Select[Prime[Range[4, 100]], Mod[#, 5] == 1 || Mod[#, 5] == 4 &]] (* _Alonso del Arte_, Nov 27 2011 *)
%o (PARI) forprime(p=2,1e3,if(kronecker(5,p)>=0,print1(p", "))) \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Magma) [ p: p in PrimesUpTo(700) | p mod 5 in {0,1,4}]; // _Vincenzo Librandi_, Aug 21 2012
%o (GAP) Filtered(Concatenation([5],Flat(List([1..140],k->[5*k-1,5*k+1]))),IsPrime); # _Muniru A Asiru_, Nov 24 2018
%Y Cf. A045468, A089270.
%Y Cf. A038872 (d=5); A038873 (d=8); A068228, A141123 (d=12); A038883 (d=13). A038889 (d=17); A141111, A141112 (d=65).
%Y Cf. A003631 (complement with respect to A000040).
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E Corrected and extended by Peter K. Pearson, May 29 2005
%E Edited by _N. J. A. Sloane_, Jul 28 2008 at the suggestion of _R. J. Mathar_
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