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

Author

Henry Bottomley, Feb 23 2000

Keywords

• nonn

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

From Jianing Song, Jun 16 2024: (Start)

Primes p such that A001176(p) = 4.

For p > 2, p is in this sequence if and only if A001175(p) == 4 (mod 8), and if and only if A001177(p) is odd. For a proof of the equivalence between A001176(p) = 4 and A001177(p) being odd, see Section 2 of my link below.

This sequence contains all primes congruent to 13, 17 (mod 20). This corresponds to case (1) for k = 3 in the Conclusion of Section 1 of my link below. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

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.
• Jianing Song, Lucas sequences and entry point modulo p
• Eric Weisstein's World of Mathematics, Lucas Number

Crossrefs

Cf. A001175, A001177.

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

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.

| m=1 | m=2 | m=3

-----------------------------+----------+---------+---------

The sequence {x(n)} | A000045 | A000129 | A006190

The sequence {w(k)} | A001176 | A214027 | A322906

Primes p such that w(p) = 1 | A112860* | A309580 | A309586

Primes p such that w(p) = 2 | A053027 | A309581 | A309587

Primes p such that w(p) = 4 | this seq | A261580 | A309588

Numbers k such that w(k) = 1 | A053031 | A309583 | A309591

Numbers k such that w(k) = 2 | A053030 | A309584 | A309592

Numbers k such that w(k) = 4 | A053029 | A309585 | A309593

* and also A053032 U {2}

Programs

Mathematica

Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; badP={}; Do[p=Prime[n]; k=1; While[k0, k++ ]; If[k==p, AppendTo[badP, p]], {n, 200}]; badP

Formula

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

Extensions

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