login
Odd primes p with one zero in Fibonacci numbers mod p.
27

%I #51 Jun 17 2024 10:49:02

%S 11,19,29,31,59,71,79,101,131,139,151,179,181,191,199,211,229,239,251,

%T 271,311,331,349,359,379,419,431,439,461,479,491,499,509,521,541,571,

%U 599,619,631,659,691,709,719,739,751,809,811,839,859,911,919,941,971

%N Odd primes p with one zero in Fibonacci numbers mod p.

%C Also, odd primes that divide Lucas numbers of odd index. - _T. D. Noe_, Jul 25 2003

%C From _Charles R Greathouse IV_, Dec 14 2016: (Start)

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

%C 35 of the first 10^2 primes

%C 330 of the first 10^3 primes

%C 3328 of the first 10^4 primes

%C 33371 of the first 10^5 primes

%C 333329 of the first 10^6 primes

%C 3333720 of the first 10^7 primes

%C 33333463 of the first 10^8 primes

%C etc. (End)

%C 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

%C Odd primes in A053031. - _Jianing Song_, Jun 19 2019

%H T. D. Noe, <a href="/A053032/b053032.txt">Table of n, a(n) for n = 1..1000</a>

%H C. Ballot and M. Elia, <a href="http://www.fq.math.ca/Papers1/45-1/quartballot01_2007.pdf">Rank and period of primes in the Fibonacci sequence; a trichotomy</a>, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B1).

%H Nicholas Bragman and Eric Rowland, <a href="https://arxiv.org/abs/2202.00704">Limiting density of the Fibonacci sequence modulo powers of p</a>, arXiv:2202.00704 [math.NT], 2022.

%H M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>

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

%e From _Michael B. Porter_, Jan 25 2019: (Start)

%e 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.

%e 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.

%e (End)

%t 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 *)

%o (PARI) fibmod(n,m)=(Mod([1, 1; 1, 0], m)^n)[1, 2]

%o 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

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

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

%Y 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.

%Y | m=1 | m=2 | m=3

%Y -----------------------------+------------+---------+---------

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

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

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

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

%Y Primes p such that w(p) = 4 | A053028*** | A261580 | A309588

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

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

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

%Y * and also this sequence U {2}

%Y ** also primes dividing Lucas numbers of even index

%Y *** also primes dividing no Lucas number

%K nonn

%O 1,1

%A _Henry Bottomley_, Feb 23 2000