Pisano period
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In mathematics, the nth Pisano period, written π(n), is the period with which the sequence of Fibonacci numbers, modulo n repeats. For example, the Fibonacci numbers mod 3 are 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, etc., with the first eight numbers repeating, so π(3) = 8.
The first Pisano periods (sequence A001175 in OEIS) for n = 1, 2, ... are:
It has been observed that zero occurs only once, twice or four times throughout any Pisano sequence. Also, it can be proven that
- π(n) ≤ 6n,
and that the equality applies infinitely often (whenever n/2 equals a power of 5).
Pisano periods are named after Leonardo Pisano, better known as Fibonacci.
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