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A003631 Primes congruent to {2, 3} mod 5.
(Formerly M0832)
37
2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 433, 443, 457, 463, 467, 487, 503, 523, 547, 557, 563, 577 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For n>1, sequence gives primes ending in 3 or 7. - Lekraj Beedassy, Oct 27 2003

Inert rational primes in Q(sqrt 5), or, p is not a square mod 5.

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

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)=8 * F(7)=13) then 8 * 13 mod 7 = 6 (p-1=6). - Louis Mello (Mellols(AT)aol.com), Feb 09 2001

These are also the primes p that divide Fibonacci(p+1) - Jud McCranie.

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

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

Primes p such that (1-x^5)/(1-x) is irreducible over GF(p). [Joerg Arndt, Aug 10 2011]

Primes p such that p does not divide sum(i=1..p-1, Fibonacci(i)^2). The sum is A001654(p-1). [Arkadiusz Wesolowski, Jul 23 2012]

The prime 2 and primes p such that p^2 mod 10 = 9.  - Richard R. Forberg, Aug 28 2013

REFERENCES

Hardy and Wright, An Introduction to the Theory of Numbers, Chap. X, p. 150, Oxford University Press, Fifth edition

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

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

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

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

MATHEMATICA

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

PROG

(Haskell)

a003631 n = a003631_list !! (n-1)

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

-- Reinhard Zumkeller, Nov 27 2012, Jul 19 2011

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

(PARI) {a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( 5, m )== -1, cnt++ )); m}; /* Michael Somos, Aug 14 2012 */

(MAGMA) [ p: p in PrimesUpTo(1000) | p mod 5 in {2, 3} ]; // Vincenzo Librandi, Aug 07 2012

CROSSREFS

Cf. A019546, A000040.

Cf. Primes in A047221.

Sequence in context: A045329 A106306 A069104 * A175443 A032449 A129941

Adjacent sequences:  A003628 A003629 A003630 * A003632 A003633 A003634

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane, Mira Bernstein

STATUS

approved

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Last modified August 27 19:27 EDT 2014. Contains 246147 sequences.