A038872 Primes congruent to {0, 1, 4} mod 5.

5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601, 619

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also odd primes p such that 5 is a square mod p: (5/p) = +1 for p > 5.

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.]

Also primes of the form 5x^2 - y^2 (cf. A031363). - N. J. A. Sloane, May 30 2014

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

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

The prime factors of A028387(n^2+3n+1). - Richard R. Forberg, Dec 12 2014

Except for p = 5, these are primes p that divide Fibonacci(p-1). - Dmitry Kamenetsky, Jul 27 2015

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

The possible prime factors of x^2 - x - 1. - Charles R Greathouse IV, Mar 18 2022

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• C. Banderier, Calcul de (5/p)
• Henri Darmon, Andrew Wiles's Marvelous Proof, Notices of the AMS (2017), Volume 64, Number 3 pp. 209-216. See p. 211.
• Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

Cf. A045468, A089270.

Cf. A038872 (d=5); A038873 (d=8); A068228, A141123 (d=12); A038883 (d=13). A038889 (d=17); A141111, A141112 (d=65).

Cf. A003631 (complement with respect to A000040).

Programs

GAP

Filtered(Concatenation([5],Flat(List([1..140],k->[5*k-1,5*k+1]))),IsPrime); # Muniru A Asiru, Nov 24 2018

Magma

[ p: p in PrimesUpTo(700) | p mod 5 in {0,1,4}]; // Vincenzo Librandi, Aug 21 2012

Maple

select(isprime, [5, seq(op([5*k-1,5*k+1]),k=1..1000)]); # Robert Israel, Dec 22 2014

Mathematica

Join[{5}, Select[Prime[Range[4, 100]], Mod[#, 5] == 1 || Mod[#, 5] == 4 &]] (* Alonso del Arte, Nov 27 2011 *)

PARI

forprime(p=2,1e3,if(kronecker(5,p)>=0,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

Formula

a(n) = A045468(n-1) for n > 1. - Robert Israel, Dec 22 2014

a(n) ~ 2n*log(n). - Charles R Greathouse IV, Nov 29 2016

Extensions

Corrected and extended by Peter K. Pearson, May 29 2005

Edited by N. J. A. Sloane, Jul 28 2008 at the suggestion of R. J. Mathar