A053027 Odd primes p with 2 zeros in Fibonacci numbers mod p.

3, 7, 23, 41, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 241, 263, 281, 283, 307, 347, 367, 383, 401, 409, 443, 449, 463, 467, 487, 503, 523, 547, 563, 569, 587, 601, 607, 641, 643, 647, 683, 727, 743, 769, 787, 823, 827, 863, 881, 883, 887, 907, 929

Offset

1,1

Comments

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

Primes in A053030. - Jianing Song, Jun 19 2019

From Jianing Song, Jun 16 2024: (Start)

Primes p such that A001176(p) = 2.

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

This sequence contains all primes congruent to 3, 7 (mod 20). This corresponds to case (2) for k = 3 in the Conclusion of Section 1 of my link below.

Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

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

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 B3).

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

M. Renault, Fibonacci sequence modulo m

Jianing Song, Entry points and periods of Lucas sequences

Formula

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

Crossrefs

Cf. A001175, A001177.

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

For a list of sequences related to the numbers of zeros in a fundamental period of {x(n)}, where {x(n)} is a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n), see A053032.

Sequence in context: A213897 A291776 A135570 * A133432 A133433 A029932

Adjacent sequences: A053024 A053025 A053026 * A053028 A053029 A053030

Keyword

nonn,changed

Author

Henry Bottomley, Feb 23 2000

Status

approved