Pisano Period

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).

REFERENCES:

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. 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.