A053028 Odd primes p with 4 zeros in any period of the Fibonacci numbers mod p.

5, 13, 17, 37, 53, 61, 73, 89, 97, 109, 113, 137, 149, 157, 173, 193, 197, 233, 257, 269, 277, 293, 313, 317, 337, 353, 373, 389, 397, 421, 433, 457, 557, 577, 593, 613, 617, 653, 661, 673, 677, 701, 733, 757, 761, 773, 797, 821, 829, 853, 857, 877, 937, 953

Offset

1,1

Comments

Also, primes that do not divide any Lucas number. - T. D. Noe, Jul 25 2003

Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. In fact, exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. The Lucas numbers separate the primes into three disjoint sets: (A053028) primes that do not divide any Lucas number, (A053027) primes that divide Lucas numbers of even index and (A053032) primes that divide Lucas numbers of odd index. - T. D. Noe, Jul 25 2003; revised by N. J. A. Sloane, Feb 21 2004

Links

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

C. Ballot and M. Elia, Rank and period of primes in the Fibonacci sequence; a trichotomy, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B2).

Nicholas Bragman and Eric Rowland, Limiting density of the Fibonacci sequence modulo powers of p, arXiv:2202.00704 [math.NT], 2022.

J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.

J. C. Lagarias, Errata to: The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 162, No. 2, (1994), 393-396.

Diego Marques and Pavel Trojovsky, The order of appearance of the product of five consecutive Lucas numbers, Tatra Mountains Math. Publ. 59 (2014), 65-77.

Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.

M. Renault, Fibonacci sequence modulo m

H. Sedaghat, Zero-Avoiding Solutions of the Fibonacci Recurrence Modulo A Prime, Fibonacci Quart. 52 (2014), no. 1, 39-45. See p. 45.

Eric Weisstein's World of Mathematics, Lucas Number

Formula

A prime p = prime(i) is in this sequence if p > 2 and A001602(i) is odd. - T. D. Noe, Jul 25 2003

Mathematica

Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; badP={}; Do[p=Prime[n]; k=1; While[k<p&&Mod[Lucas[k], p]>0, k++ ]; If[k==p, AppendTo[badP, p]], {n, 200}]; badP

Crossrefs

Cf. A001176.

Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)), A053027, A053032.

Sequence in context: A191108 A216575 A306626 * A189411 A248980 A188131

Adjacent sequences: A053025 A053026 A053027 * A053029 A053030 A053031

Keyword

nonn

Author

Henry Bottomley, Feb 23 2000

Extensions

Edited: Name clarified. Moree and Renault link updated. Ballot and Elia reference linked. - Wolfdieter Lang, Jan 20 2015

Status

approved