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Pisano Period


The sequence of Fibonacci numbers {F_n} is periodic modulo any modulus m (Wall 1960), and the period (mod m) is the known as the Pisano period pi(m) (Wrench 1969). For m=1, 2, ..., the values of pi(m) are 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, ... (OEIS A001175).

Since pi(10)=60, the last digit of F_n 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 15000 and the last five a period of 150000. Jarden subsequently showed that for d>=3, the last d digits have a period of 15·10^(d-1) (Livio 2002, pp. 105-106). The sequence of Pisano periods for n=1, 10, 100, 1000, ... are therefore 60, 300, 1500, 15000, 150000, 1500000, ... (OEIS A096363).

pi(m) is even if m>2 (Wall 1960). pi(m)=m iff m=24·5^(k-1) for some integer k>1 (Fulton and Morris 1969, Wrench 1969).


See also

Fibonacci Number

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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 m." 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.

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Pisano Period

Cite this as:

Weisstein, Eric W. "Pisano Period." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PisanoPeriod.html

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