A053032 Odd primes p with one zero in Fibonacci numbers mod p.

11, 19, 29, 31, 59, 71, 79, 101, 131, 139, 151, 179, 181, 191, 199, 211, 229, 239, 251, 271, 311, 331, 349, 359, 379, 419, 431, 439, 461, 479, 491, 499, 509, 521, 541, 571, 599, 619, 631, 659, 691, 709, 719, 739, 751, 809, 811, 839, 859, 911, 919, 941, 971

Offset

1,1

Comments

Also, odd primes that divide Lucas numbers of odd index. - T. D. Noe, Jul 25 2003

From Charles R Greathouse IV, Dec 14 2016: (Start)

It seems that this sequence contains about 1/3 of the primes. In particular, members of this sequence constitute:

35 of the first 10^2 primes

330 of the first 10^3 primes

3328 of the first 10^4 primes

33371 of the first 10^5 primes

333329 of the first 10^6 primes

3333720 of the first 10^7 primes

33333463 of the first 10^8 primes

etc. (End)

Of the Fibonacci-like sequences modulo a prime p that are not A000004, one of them has a period length less than A001175(p) if and only if p = 5 or p is in this sequence. - Isaac Saffold, Dec 18 2018

Odd primes in A053031. - Jianing Song, Jun 19 2019

References

Ballot, Christian. "Prime Factors of Fibonacci-Related Recurrences." The Fibonacci Quarterly 63.2 (2025): 178-206.

Links

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

Christian Ballot and Michele Elia, Rank and period of primes in the Fibonacci sequence; a trichotomy, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B1).

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

Marc T. Pudelko, Modular Periodicity of Random Initialized Recurrences, arXiv:2510.24882 [math.NT], 2025. See pp. 6, 9.

Marc Renault, Fibonacci sequence modulo m.

Formula

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

Example

From Michael B. Porter, Jan 25 2019: (Start)
The Fibonacci numbers (mod 7) repeat the pattern 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1. Since there are two zeros, 7 is not in the sequence.
The Fibonacci numbers (mod 11) repeat the pattern 0, 1, 1, 2, 3, 5, 8, 2, 10, 1 which has only one zero, so 11 is in the sequence.
(End)

Mathematica

Prime@ Rest@ Position[Table[Count[Drop[NestWhile[Append[#, Mod[Total@ Take[#, -2], n]] &, {1, 1}, If[Length@ # < 3, True, Take[#, -2] != {1, 1}] &], -2], 0], {n, Prime@ Range@ 168}], 1][[All, 1]] (* Michael De Vlieger, Aug 08 2018 *)

Prog

(PARI) fibmod(n, m)=(Mod([1, 1; 1, 0], m)^n)[1, 2]
is(n)=my(k=n+[0, -1, 1, 1, -1][n%5+1]); k>>=valuation(k, 2)-1; fibmod(k, n)==0 && fibmod(k/2, n) && isprime(n) \\ Charles R Greathouse IV, Dec 14 2016

Crossrefs

Cf. A001175, A001177. See A112860 for another version.

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 | A053028*** | 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 this sequence U {2}

** also primes dividing Lucas numbers of even index

*** also primes dividing no Lucas number

Sequence in context: A045468 A196095 A268271 * A342961 A277123 A034099

Adjacent sequences: A053029 A053030 A053031 * A053033 A053034 A053035

Keyword

nonn,changed

Author

Henry Bottomley, Feb 23 2000

Status

approved