|
%I M0961 N0358 #81 Apr 03 2023 10:36:09
%S 0,2,4,5,7,8,11,13,16,17,19,31,37,41,47,53,61,71,79,113,313,353,503,
%T 613,617,863,1097,1361,4787,4793,5851,7741,8467,10691,12251,13963,
%U 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
%N Indices of prime Lucas numbers.
%C Some of the larger entries may only correspond to probable primes.
%C 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
%C 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
%C 2316773 is in the sequence, but its position is not yet defined. L(2316773) is a 484177-digit PRP. - _Serge Batalov_, Jun 11 2017
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D S. Vajda, Fibonacci and Lucas numbers and the Golden Section: Theory and Applications. Chichester: Ellis Horwood Ltd., 1989.
%H J. Brillhart, P. L. Montgomery and R. D. Silverman, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0917832-6">Tables of Fibonacci and Lucas factorizations</a>, Math. Comp. 50 (1988), 251-260.
%H D. Broadhurst, <a href="http://groups.yahoo.com/group/primenumbers/message/1003">Lucas record follows Fibonacci</a>, Yahoo! group "primenumbers", Apr 26 2001
%H D. Broadhurst, <a href="/A001606/a001606.png">Lucas record follows Fibonacci</a> [Cached copy]
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=LucasPrime">Lucas prime</a>
%H H. Dubner and W. Keller, <a href="http://dx.doi.org/10.1090/S0025-5718-99-00981-3">New Fibonacci and Lucas Primes</a>, Math. Comp. 68 (1999) 417-427.
%H Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.
%H Blair Kelly, <a href="http://mersennus.net/fibonacci//lucas.txt">Factorizations of Lucas numbers</a>
%H Alex Kontorovich and Jeff Lagarias, <a href="https://arxiv.org/abs/1808.03235">On Toric Orbits in the Affine Sieve</a>, arXiv:1808.03235 [math.NT], 2018.
%H H. Lifchitz and R. Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=L(n)">PRP Top Records, L(n)</a>
%H Mersenneforum, <a href="http://www.mersenneforum.org/showthread.php?p=459964#post459964">A collection of factors of L(2^m)</a>.
%H Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
%H The Prime Database, <a href="https://t5k.org/primes/page.php?id=115420"> V(81671)</a>
%H Lawrence Somer and Michal Křížek, <a href="https://www.fq.math.ca/Papers1/53-1/SomerKrizek5222014.pdf">On Primes in Lucas Sequences</a>, Fibonacci Quart. 53 (2015), no. 1, 2-23.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>.
%t Reap[For[k = 0, k < 20000, k++, If[PrimeQ[LucasL[k]], Print[k]; Sow[k]]] ][[2, 1]] (* _Jean-François Alcover_, Feb 27 2016 *)
%o (PARI) is(n)=ispseudoprime(fibonacci(n-1)+fibonacci(n+1)) \\ _Charles R Greathouse IV_, Apr 24 2015
%Y Cf. A000032, A000204, A001605, A005479.
%Y Cf. A080327 (n for which Lucas(n) and Fibonacci(n) are both prime).
%K nonn,hard,nice
%O 1,2
%A _N. J. A. Sloane_
%E 4 more terms from _David Broadhurst_, Jun 08 2001
%E More terms from _T. D. Noe_, Feb 15 2003 and Mar 04 2003; see link to The Prime Glossary.
%E 387433, 443609, 532277 and 574219 found by Renaud Lifchitz, contributed by _Eric W. Weisstein_, Nov 29 2005
%E 616787, 631181, 637751, 651821, 692147 found by _Henri Lifchitz_, circa Oct 01 2008, contributed by _Alexander Adamchuk_, Nov 28 2008
%E 901657 and 1051849 found by Renaud Lifchitz, circa Nov 2008 and Mar 2009, contributed by _Alexander Adamchuk_, May 15 2010
%E 1 more term from _Serge Batalov_, Jun 11 2017
|
