The Pisano period
is the least positive period of the sequence of Fibonacci
numbers modulo a positive integer (Wall 1960, Wrench 1969). For , 2, ..., the values of are 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, ... (OEIS
A001175).
Since ,
the last digit of
repeats with period 60, as first noted by Lagrange in 1774 (Livio 2002, p. 105).
The last two digits repeat with a period of 300, and the last three with a period
of 1500. In 1963, Geller found that the last four digits have a period of and the last five a period of . Jarden subsequently showed that for , the last digits have a period of (Livio 2002, pp. 105-106). The sequence
of Pisano periods for ,
100, 1000, 10000, ... is therefore 60, 300, 1500, 15000, 150000, 1500000, ... (OEIS
A096363).
is even if (Wall 1960). For , iff for some nonnegative
integer
(Fulton and Morris 1969, Wrench 1969).
Fulton, J. D. and Morris, W. L. "On Arithmetical Functions Related to the Fibonacci Numbers." Acta Arith.16, 105-110,
1969. https://doi.org/10.4064/aa-16-2-105-110.Hannon,
B. H. and Morris, W. L. Tables
of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261,
Oak Ridge National Laboratory, Oak Ridge, Tennessee, June 1968.Livio,
M. The
Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New
York: Broadway Books, 2002.Numberphile. "Fibonacci Mystery."
Sep. 18, 2013. https://www.youtube.com/watch?v=Nu-lW-Ifyec.Reiter,
C. A. "Fibonacci Numbers: Reduction Formulas and Short Periods." Fib.
Quart.31, 315-324, 1993.Roberts, J. The
Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 162, 1992.Sato,
N. (Ed.). "Mathematical Mayhem. Shreds and Slices: Fibonacci Residues."
Crux Math.23, 224-226, 1997.Sloane, N. J. A.
Sequences A001175/M2710 and A096363
in "The On-Line Encyclopedia of Integer Sequences."Wall, D. D.
"Fibonacci Series Modulo ." Amer. Math. Monthly67, 525-532, 1960.Wrench,
J. W. "Review of B. H. Hannon and W. L. Morris, Tables
of Arithmetical Functions Related to the Fibonacci Numbers." Math. Comput.23,
459-460, 1969.
is the least positive period of the sequence of Fibonacci
numbers modulo a positive integer 
(Wall 1960, Wrench 1969). For 
, 2, ..., the values of 
are 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, ... (OEIS
A001175).
,
the last digit of 
repeats with period 60, as first noted by Lagrange in 1774 (Livio 2002, p. 105).
The last two digits repeat with a period of 300, and the last three with a period
of 1500. In 1963, Geller found that the last four digits have a period of 
and the last five a period of 
. Jarden subsequently showed that for 
, the last 
digits have a period of 
(Livio 2002, pp. 105-106). The sequence
of Pisano periods for 
,
100, 1000, 10000, ... is therefore 60, 300, 1500, 15000, 150000, 1500000, ... (OEIS
A096363).
is even if 
(Wall 1960). For 
, 
iff 
for some nonnegative
integer 
(Fulton and Morris 1969, Wrench 1969).
." Amer. Math. Monthly 67, 525-532, 1960.Wrench,
J. W. "Review of B. H. Hannon and W. L. Morris, Tables
of Arithmetical Functions Related to the Fibonacci Numbers." Math. Comput. 23,
459-460, 1969.