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A003631
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Primes congruent to {2, 3} mod 5.
(Formerly M0832)
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36
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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
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OFFSET
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1,1
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COMMENTS
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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]
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REFERENCES
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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MATHEMATICA
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Select[ Prime[Range[106]], MemberQ[{2, 3}, Mod[#, 5]] &] (* Robert G. Wilson v, Sep 12 2011 *)
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PROG
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(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
(MAGMA) [ p: p in PrimesUpTo(1000) | p mod 5 in {2, 3} ]; // Vincenzo Librandi, Aug 07 2012
(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 */
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CROSSREFS
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Cf. A019546, A000040.
Cf. Primes in A047221.
Sequence in context: A045329 A106306 A069104 * A175443 A032449 A129941
Adjacent sequences: A003628 A003629 A003630 * A003632 A003633 A003634
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Mira Bernstein
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STATUS
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approved
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