A001606 Indices of prime Lucas numbers.

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

Comments

Some of the larger entries may only correspond to probable primes.

Since (as noted under A000032) L(n) divides L(mn) whenever m is odd, L(n) cannot be prime unless n is itself prime, or else n contains no odd divisor, i.e., is a power of 2. Potential divisors of L(n) must satisfy certain linear forms dependent upon the parity of n, as shown in Vajda (1989), p. 82 (with a slight typographical error in the proof). - John Blythe Dobson, Oct 22 2007

Powers of 2 in this sequence are 2, 4, 8, 16; for 5 <= m <= 24, L(2^m) is composite; no factors of L(2^m) are known for m = 25, 26, 27, 29, 32, 33... (See Link section). - Serge Batalov, May 30 2017

2316773 is in the sequence, but its position is not yet defined. L(2316773) is a 484177-digit PRP. - Serge Batalov, Jun 11 2017

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 246.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• S. Vajda, Fibonacci and Lucas numbers and the Golden Section: Theory and Applications. Chichester: Ellis Horwood Ltd., 1989.

Links

• J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
• David Broadhurst, Lucas record follows Fibonacci, Yahoo! group "primenumbers", Apr 26 2001.
• David Broadhurst, Lucas record follows Fibonacci [Cached copy].
• C. K. Caldwell, The Prime Glossary, Lucas prime.
• H. Dubner and W. Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.
• Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.
• Blair Kelly, Factorizations of Lucas numbers
• Alex Kontorovich and Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
• H. Lifchitz and R. Lifchitz, PRP Top Records, L(n).
• Mersenneforum, A collection of factors of L(2^m).
• Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
• The Prime Database, V(81671), submitted Sep. 7, 2013, updated May 20, 2023.
• Lawrence Somer and Michal Křížek, On Primes in Lucas Sequences, Fibonacci Quart. 53 (2015), no. 1, 2-23.
• Eric Weisstein's World of Mathematics, Lucas Number.
• Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Crossrefs

Cf. A000032, A000204, A001605, A005479.

Cf. A080327 (n for which Lucas(n) and Fibonacci(n) are both prime).

Subsequence of A076697 (indices for which gpf(A000032(n)) sets a new record).

Programs

Mathematica

Reap[For[k = 0, k < 20000, k++, If[PrimeQ[LucasL[k]], Print[k]; Sow[k]]] ][[2, 1]] (* Jean-François Alcover, Feb 27 2016 *)

PARI

is(n)=ispseudoprime(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Apr 24 2015

Extensions

4 more terms from David Broadhurst, Jun 08 2001

More terms from T. D. Noe, Feb 15 2003 and Mar 04 2003; see link to The Prime Glossary.

387433, 443609, 532277 and 574219 found by Renaud Lifchitz, contributed by Eric W. Weisstein, Nov 29 2005

616787, 631181, 637751, 651821, 692147 found by Henri Lifchitz, circa Oct 01 2008, contributed by Alexander Adamchuk, Nov 28 2008

901657 and 1051849 found by Renaud Lifchitz, circa Nov 2008 and Mar 2009, contributed by Alexander Adamchuk, May 15 2010

1 more term from Serge Batalov, Jun 11 2017