A296240 -id:A296240 - cp's OEIS frontend

A001175 Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n.

1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, 16, 30, 48, 24, 100, 84, 72, 48, 14, 120, 30, 48, 40, 36, 80, 24, 76, 18, 56, 60, 40, 48, 88, 30, 120, 48, 32, 24, 112, 300, 72, 84, 108, 72, 20, 48, 72, 42, 58, 120, 60, 30, 48, 96, 140, 120, 136

Offset: 1

N. J. A. Sloane

These numbers might also have been called Fibonacci periods.
Also, number of perfect multi-Skolem type sequences of order n.
Index the Fibonacci numbers so that 3 is the fourth number. If the modulo base is a Fibonacci number (>=3) with an even index, the period is twice the index. If the base is a Fibonacci number (>=5) with an odd index, the period is 4 times the index. - Kerry Mitchell, Dec 11 2005
Each row of the image represents a different modulo base n, from 1 at the bottom to 24 at the top.  The columns represent the Fibonacci numbers mod n, from 0 mod n at the left to 59 mod n on the right.  In each cell, the brightness indicates the value of the residual, from dark for 0 to near-white for n-1.  Blue squares on the left represent the first period; the number of blue squares is the Pisano number. - Kerry Mitchell, Feb 02 2013
a(n) = least positive integer k such that F(k) == 0 (mod n) and F(k+1) == 1 (mod n), where F = A000045 is the Fibonacci sequence. a(n) exists for all n by Dirichlet's box principle and the fact that the positive integers are well-ordered. Cf. Saha and Karthik (2011). - L. Edson Jeffery, Feb 12 2014

			For n=4, take the Fibonacci sequence (A000045), 1, 1, 2, 3, 5, 8, 13, 21, ... (mod 4), which gives 1, 1, 2, 3, 1, 0, 1, 1, .... This repeats a pattern of length 6, so a(4) = 6. - _Michael B. Porter_, Jul 19 2016

B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 162.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, pp. 304 - 309.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989. See p. 89. - N. J. A. Sloane, Feb 20 2013

T. D. Noe, Table of n, a(n) for n = 1..10000
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and Journal of Integer Sequences 17 (2014), #14.8.5.
Michael Baake, Natascha Neumarker, and John A. G. Roberts, Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices, arXiv:1205.1003 [math.DS], 2012.
Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.
K. S. Brown, Periods of Fibonacci Sequences mod m.
K. S. Brown, Periods of Fibonacci Sequences mod m. [Cached copy in pdf format]
Francis N. Castro and Luis A. Medina, Modular periodicity of exponential sums of symmetric Boolean functions and some of its consequences, arXiv:1603.00534 [math.NT], 2016.
D. A. Coleman, C. J. Dugan, R. A. McEwen, C. A. Reiter, and T. T. Tang, Periods of (q,r)-Fibonacci sequences and Elliptic Curves, Fibonacci Quart. 44 (1) (2006), 59-70.
Joseph Louis de Lagrange, Additions aux éléments d'algèbre d'Euler. Analyse indéterminée. (1774), pp. 143ff.
H. T. Engstrom, On sequences defined by linear recurrence relations, Trans. Am. Math. Soc. 33 (1) (1931), 210-218.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica 16 (1969), 105-110.
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, arXiv preprint arXiv:1203.4066 [math.NT], 2012.
James Grime and Brady Haran, Fibonacci Mystery, Numberphile video, 2013.
B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. [Annotated and scanned copy]
Naoki Sato, Cyrus Hsia, Richard Hoshino, Wai Ling Yee, and Adrian Chan, editors, Fibonacci residues, Crux Mathematicorum 23:4 (1997), pp. 224-226.
Dan Ismailescu and Peter C. Shim, On numbers that cannot be expressed as a plus-minus weighted sum of a Fibonacci number and a prime, INTEGERS 14 (2014), #A65.
Kerry Mitchell, Illustration of Pisano Numbers as Periods of Fibonacci Numbers Mod n
G. Nordh, Perfect Skolem sequences, arXiv:math/0506155 [math.CO], 2005.
Noel Patson, Pisano period and permutations of n X n matrices, Australian Math. Soc. Gazette, 2007.
Noel Patson, Square Matrix Permutations.
M. Renault, Periods of Fibonacci Sequence Modulo m.
Arpan Saha and C. S. Karthik, A few equivalences of [the] Wall-Sun-Sun prime conjecture, p. 2, arXiv:1102.1636 [math.NT], 2011.
Minjia Shi, Zhongyi Zhang, and Patrick Solé, Pisano period codes, arXiv:1709.04582 [cs.IT], 2017.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Eric Weisstein's World of Mathematics, Pisano Number.
Wikipedia, Pisano period.
J. W. W. [J. W. Wrench], Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460.

Cf. A060305 (Fibonacci period mod prime(n)), A003893.

Cf. A001178 (Fibonacci frequency), A001179 (Leonardo logarithm), A235702 (fixed points), A066853 (number of elements in the set of residua), A222413, A296240 (Pisano quotients for primes).

a001175 1 = 1
a001175 n = f 1 ps 0 where
   f 0 (1 : xs) pi = pi
   f _ (x : xs) pi = f x xs (pi + 1)
   ps = 1 : 1 : zipWith (\u v -> (u + v) `mod` n) (tail ps) ps
-- Reinhard Zumkeller, Jan 15 2014

a:= proc(n) local f, k, l; l:= ifactors(n)[2];
      if nops(l)<>1 then ilcm(seq(a(i[1]^i[2]), i=l))
    else f:= [0, 1];
         for k do f:=[f[2], f[1]+f[2] mod n];
                  if f=[0, 1] then break fi
         od; k
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 18 2013

Table[a={1, 0}; a0=a; k=0; While[k++; s=Mod[Plus@@a, n]; a=RotateLeft[a]; a[[2]]=s; a!=a0]; k, {n, 2, 100}] (* T. D. Noe, Jul 19 2005 *)
a[1]=1; a[n_]:= For[k = 1, True, k++, If[Mod[Fibonacci[k], n]==0 && Mod[ Fibonacci[k+1], n]==1, Return[k]]]; Table[a[n], {n, 100}] (* Jean-François Alcover, Feb 11 2015 *)
test[{0, 1, }]:= False; test[]:= True;
nest[k_][{a_, b_, c_}]:= {Mod[b, k], Mod[a+b, k], c+1};
A001175[1] := 1;
A001175[k_]:= NestWhile[nest[k], {1, 1, 1}, test][[3]];
Table[A001175[n], {n, 100}] (* Leo C. Stein, Nov 08 2019 *)
Module[{nn=1000,fibs},fibs=Fibonacci[Range[nn]];Table[Length[ FindTransientRepeat[ Mod[fibs,n],2][[2]]],{n,70}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 02 2020 *)

minWSS=2^64; \\ PrimeGrid search
fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
entryp(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1], for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k
entry(n)=if(n==1,return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i,1]>=minWSS, entryp(f[i,1]^f[i,2]), entryp(f[i,1])*f[i,1]^(f[i,2] - 1))); if(f[1,1]==2&&f[1,2]>1, v[1]=3<Charles R Greathouse IV, Feb 13 2014; updated Dec 14 2016; updated Aug 24 2021; updated Jul 08 2024

from functools import reduce
from sympy import factorint, lcm
def A001175(n):
    if n == 1:
        return 1
    f = factorint(n)
    if len(f) > 1:
        return reduce(lcm, (A001175(a**f[a]) for a in f))
    else:
        k,x = 1, [1,1]
        while x != [0,1]:
            k += 1
            x = [x[1], (x[0]+x[1]) % n]
        return k # Chai Wah Wu, Jul 17 2019

def a(n): return BinaryRecurrenceSequence(1,1).period(n) # Ralf Stephan, Jan 23 2014

Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)). - T. D. Noe, May 02 2005
a(n) = n-1 if n is a prime > 5 included in A003147 ( n = 11, 19, 31, 41, 59, 61, 71, 79, 109...). - Benoit Cloitre, Jun 04 2002
K. S. Brown shows that a(n)/n <= 6 for all n and a(n)=6n if and only if n has the form 2*5^k.
a(n) = A001177(n)*A001176(n) for n >= 1. - Henry Bottomley, Dec 19 2001
a(n) <= 2*n+2 if n is a prime > 5, by Wall's Theorems 6 and 7; see A060305, A222413, A296240. - Jonathan Sondow, Dec 10 2017
a(2^k) = 3*2^(k-1) for k > 0. In general, if a(p) != a(p^2) for p prime, then a(p^k) = p^(k-1)*a(p) [Wall, 1960]. - Chai Wah Wu, Feb 25 2022

A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

Original entry on oeis.org

211, 281, 421, 461, 521, 691, 881, 991, 1031, 1151, 1511, 1601, 1871, 1951, 2221, 2591, 3001, 3251, 3571, 3851, 4021, 4391, 4441, 4481, 4621, 4651, 4691, 4751, 4871, 5081, 5281, 5381, 5531, 5591, 5641, 5801, 5881, 6011, 6101, 6211, 6271, 6491, 6841

Offset: 1

N. J. A. Sloane

From A.H.M. Smeets, Nov 15 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 20*log(p) (see E. Lehmer).
In E. Lehmer, Artiads characterized, she counted in the table on p. 122 the primes p for which p == 1 (mod 5) instead of all primes. As a result, in the corollary on p. 121, the 20% becomes 5% (or 1/20 instead of 1/5). (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..24903 (first 1000 terms from T. D. Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138 (annotated scanned copy).
Michael J. Mossinghoff and Christopher Pinner, Prime power order circulant determinants, arXiv:2205.12439 [math.NT], 2022. See Type 2 primes on p. 3.
H. W. Lloyd Tanner, On the Binomial Equation x^p-1=0: Quinquisection, Proc. London Math. Soc., 18 (1886-1887), 214-234.
H. W. Lloyd Tanner, On Complex Primes formed with the Fifth Roots of Unity, Proc. London Math. Soc., 24 (1892-1893), 223-262.

Cf. A047650, A000045, A024894, subsequence of A030430.

See also A270798 (a subsequence), A270800.

a001583 n = a001583_list !! (n-1)
a001583_list = filter
   (\p -> mod (a000045 $ div (p - 1) 5) p == 0) a030430_list
-- Reinhard Zumkeller, Aug 15 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* Jean-François Alcover, Jun 22 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
list(lim)=my(v=List()); forprime(p=11,lim, if(p%5==1 && fibmod(p\5,p)==0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 06 2017

From A.H.M. Smeets, Nov 15 2023: (Start)
Equals {prime(m): A296240(m) == 0 (mod 5)}.
a(n) ~ prime(20*n). (End)

More terms from James Sellers, Jan 25 2000

Edited by N. J. A. Sloane, Apr 01 2016

A060305 Pisano periods for primes: period of Fibonacci numbers mod prime(n).

Original entry on oeis.org

3, 8, 20, 16, 10, 28, 36, 18, 48, 14, 30, 76, 40, 88, 32, 108, 58, 60, 136, 70, 148, 78, 168, 44, 196, 50, 208, 72, 108, 76, 256, 130, 276, 46, 148, 50, 316, 328, 336, 348, 178, 90, 190, 388, 396, 22, 42, 448, 456, 114, 52, 238, 240, 250, 516, 176, 268, 270, 556

Offset: 1

Louis Mello (mellols(AT)aol.com), Mar 26 2001

nonn

Assuming Wall's conjecture (which is still open) allows one to calculate A001175(m) when m is a prime power since for any k >= 1: A001175(prime(n)^k) = a(n)*prime(n)^(k-1). For example: A001175(2^k) = 3*2^(k-1) = A007283(k-1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014), # 14.8.5.
A. Elsenhans and J. Jahnel, The Fibonacci sequence modulo p^2 -- An investigation by computer for p < 10^14, (2004).
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Wikipedia, Pisano period.

Cf. A001175, A000961, A071774, A003147, A296240.

a:= proc(n) option remember; local F, k, p;
      F:=[1,1]; p:=ithprime(n);
      for k while F<>[0,1] do
        F:=[F[2], irem(F[1]+F[2],p)]
      od: k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 16 2015

Table[p=Prime[n]; a={1,0}; a0=a; k=0; While[k++; s=Mod[Plus@@a,p];a=RotateLeft[a]; a[[2]]=s; a!=a0]; k, {n,100}] (* T. D. Noe, Jun 12 2006 *)

for(n=1,100,s=1; while(sum(i=n,n+s,abs(fibonacci(i)%prime(n)-fibonacci(i+s)%prime(n)))+sum(i=n+1,n+1+s,abs(fibonacci(i)%prime(n)-fibonacci(i+s)%prime(n)))>0,s++); print1(s,","))

from itertools import count
from sympy import prime
def A060305(n):
    x, p = (1,1), prime(n)
    for k in count(1):
        if x == (0,1):
            return k
        x = (x[1], (x[0]+x[1]) % p) # Chai Wah Wu, May 31 2022

a(n) = A001175(prime(n)). - Jonathan Sondow, Dec 09 2017

a(n) = (3 - L(p))/2 * (p - L(p)) / A296240(n) for n >= 4, where p = prime(n) and L(p) = Legendre(p|5); so a(n) <= p-1 if p == +- 1 mod 5, and a(n) <= 2*p+2 if p == +- 2 mod 5. See Wall's Theorems 6 and 7. - Jonathan Sondow, Dec 10 2017

Corrected by Benoit Cloitre, Jun 04 2002

Name clarified by Jonathan Sondow, Dec 09 2017

A047650 Primes for which golden mean tau is a quadratic residue.

Original entry on oeis.org

29, 89, 101, 181, 229, 349, 401, 461, 509, 521, 541, 709, 761, 769, 809, 941, 1009, 1021, 1049, 1061, 1109, 1229, 1249, 1289, 1361, 1409, 1549, 1601, 1621, 1669, 1709, 1721, 1741, 1789, 1861, 2029, 2069, 2081, 2089, 2389, 2441, 2621, 2729, 2801, 2861

Offset: 0

N. J. A. Sloane

Primes of the form x^2 + 20*y^2. - T. D. Noe, May 08 2005
Also primes p that divide the sum of cubes of the first (p-1)/2 Fibonacci numbers A005968((p-1)/2). - Alexander Adamchuk, Aug 07 2006
From A.H.M. Smeets, Nov 16 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 8*log(p).
In x^2 + 20y^2: x == 1 (mod 2) and x !== 5 (mod 10). Otherwise not prime. (End)

A.H.M. Smeets, Table of n, a(n) for n = 0..20000 (terms 0..1700 from Vincenzo Librandi)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138.
E. Lehmer, Correction, Fib. Quart., 4 (1966), 135-138.
E. Lehmer, On the quadratic character of the Fibonacci root (annotated scanned copy)

Cf. A047651, A001583.

Cf. A005968.

k:=20; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Vincenzo Librandi, Sep 05 2016

nn=20; pMax=3000; Union[Reap[Do[p=x^2 + nn*y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]] (* Vincenzo Librandi, Sep 05 2016 *)

From A.H.M. Smeets, Nov 16 2023: (Start)
Equals {prime(n): A296240(n) in {2^k: k > 0}} = {A308787} union {A308789} union {A308793} union ... .
a(n) ~ A000040(8*n). (End)

More terms from James Sellers, Jan 25 2000

cp's OEIS Frontend

A001175 Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n.

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A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

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A060305 Pisano periods for primes: period of Fibonacci numbers mod prime(n).

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A047650 Primes for which golden mean tau is a quadratic residue.

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A222413 All primes p > 5 such that A001175(p) is smaller than the maximal value permitted by Wall's Theorems 6 and 7.

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A339855 Primes p such that the absolute value of the fraction A241014(A000720(p)) / p is a record low.

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A347565 Primes p such that A241014(A000720(p)) is +1 or -1.

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A366951 a(n) = 2(p_n - 1)/A060305(n) iff p_n == +/- 1 (mod 5), 2(p_n + 1)/A060305(n) iff p_n == +/- 2 (mod 5), 0 iff p_n = 5.