The sequence of Fibonacci numbers is periodic modulo any modulus (Wall 1960), and the period (mod ) is the known as the Pisano period (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 ,
10, 100, 1000, ... are therefore 60, 300, 1500, 15000, 150000, 1500000, ... (OEIS
A096363).
is even if (Wall 1960). iff for some 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.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.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 periodic modulo any modulus 
(Wall 1960), and the period (mod 
) is the known as the Pisano period 
(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 
,
10, 100, 1000, ... are therefore 60, 300, 1500, 15000, 150000, 1500000, ... (OEIS
A096363).
is even if 
(Wall 1960). 
iff 
for some 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.