%I M0832 #115 Oct 07 2023 16:21:36

%S 2,3,7,13,17,23,37,43,47,53,67,73,83,97,103,107,113,127,137,157,163,

%T 167,173,193,197,223,227,233,257,263,277,283,293,307,313,317,337,347,

%U 353,367,373,383,397,433,443,457,463,467,487,503,523,547,557,563,577

%N Primes congruent to 2 or 3 modulo 5.

%C For n>1, sequence gives primes ending in 3 or 7. - _Lekraj Beedassy_, Oct 27 2003

%C Inert rational primes in Q(sqrt 5), or, p is not a square mod 5. [See e.g., Hasse, Legendre symbol (5|p) = -1, Hardy and Wright, Theorem 257 (2), p. 222, and Dodd Appendix B, pp. 128 - 150, primes p < 32771 with (p,0). - _Wolfdieter Lang_, Jun 16 2021]

%C Primes for which the period of the Fibonacci sequence mod p divides 2p+2.

%C Let F(n) be the n-th Fibonacci number for n=1,2,3,... (A000045). F(n) mod p (a prime) generates a periodic sequence. This sequence may be generated as follows: F(p-1)* F(p) mod p = p-1. E.g., p=7: F(6) * F(7) mod 7 = 8 * 13 mod 7 = 6 = p-1. - Louis Mello (Mellols(AT)aol.com), Feb 09 2001

%C These are also the primes p that divide Fibonacci(p+1). - _Jud McCranie_

%C Also primes p such that p divides F(2p+1)-1; such that p divides F(2p+3)-1; such that p divides F(3p+1)-1. - _Benoit Cloitre_, Sep 05 2003

%C Primes p such that the polynomial x^2-x-1 mod p has no zeros; i.e., x^2-x-1 is irreducible over the integers mod p. - _T. D. Noe_, May 02 2005

%C Primes p such that (1-x^5)/(1-x) is irreducible over GF(p). - _Joerg Arndt_, Aug 10 2011

%C Primes p such that p does not divide Sum_{i=1..p-1} Fibonacci(i)^2 = A001654(p-1). - _Arkadiusz Wesolowski_, Jul 23 2012

%C The prime 2 and primes p such that p^2 mod 10 = 9. - _Richard R. Forberg_, Aug 28 2013

%C Primes p such that 5 divides sigma(p^3), cf. A274397. - _M. F. Hasler_, Jul 10 2016

%D F. W. Dodd, Number Theory in the Quadratic Field with Golden Section Unit, Polygon Publishing House, Passaic, NJ 07055, 1983, Appendix B, pp. 128 - 150.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chap. X, p. 150, Chap. XV, Theorem 257 (2), p. 222, Oxford University Press, Fifth edition.

%D H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D N. N. Vorob'ev, Fibonacci Numbers, Pergamon Press, 1961.

%H T. D. Noe, <a href="/A003631/b003631.txt">Table of n, a(n) for n = 1..1000</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 <a href="https://oeis.org/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>

%F a(n) ~ 2n log n. - _Charles R Greathouse IV_, Jun 19 2017

%t Select[ Prime[Range[106]], MemberQ[{2, 3}, Mod[#, 5]] &] (* _Robert G. Wilson v_, Sep 12 2011 *)

%t a[ n_] := If[ n < 1, 0, Module[{c = 0, m = 0}, While[ c < n, If[ PrimeQ[++m] && KroneckerSymbol[5, m] == -1, c++]]; m]]; (* _Michael Somos_, Nov 24 2018 *)

%o (Haskell)

%o a003631 n = a003631_list !! (n-1)

%o a003631_list = filter ((== 1) . a010051') a047221_list

%o -- _Reinhard Zumkeller_, Nov 27 2012, Jul 19 2011

%o (PARI) list(lim)=select(n->n%5==2||n%5==3,primes(primepi(lim))) \\ _Charles R Greathouse IV_, Jul 25 2011

%o (PARI) {a(n) = if( n < 1, 0, my(c ,m); while( c < n, if( isprime(m++) && kronecker(5, m) == -1, c++)); m)}; /* _Michael Somos_, Aug 14 2012 */

%o (Magma) [ p: p in PrimesUpTo(1000) | p mod 5 in {2, 3} ]; // _Vincenzo Librandi_, Aug 07 2012

%Y Primes in A047221.

%Y Cf. A000040.

%Y Cf. A000045, A001654.

%Y Cf. A274397.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_, _Mira Bernstein_