

A038872


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


75



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
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OFFSET

1,1


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 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
Except for p = 5, these are primes p that divide Fibonacci(p1).  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


REFERENCES

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


LINKS



FORMULA



MAPLE

select(isprime, [5, seq(op([5*k1, 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 *)


PROG

(Magma) [ p: p in PrimesUpTo(700)  p mod 5 in {0, 1, 4}]; // Vincenzo Librandi, Aug 21 2012
(GAP) Filtered(Concatenation([5], Flat(List([1..140], k>[5*k1, 5*k+1]))), IsPrime); # Muniru A Asiru, Nov 24 2018


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

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


STATUS

approved



