Periodic Fibonacci

Number Theory Level 3

Consider the construction of a new sequence S S by taking the Fibonacci sequence modulo 10.

The first few terms of S S are S = 0 , 1 , 1 , 2 , 3 , 5 , 8 , 3 , 1 , 4 , 5 , 9 , … S = 0,1,1,2,3,5,8,3,1,4,5,9,\ldots .

Find the fundamental period of S S .

If S S is not period, enter your answer as -1.


The answer is 60.

Ossama Ismail by Ossama Ismail , 63, Egypt

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1 solution

Michael Mendrin
Feb 9, 2017

The is the Pisano number, π ( 10 ) = 60 \pi (10)=60 . See Fibonacc 0 0 ... for more about this.

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For those of you who like coding : 

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int get pisano period( m) {   // m = 10

    int  a = 0, b = 1, c = a + b;

    for (int i = 0; i < m * m; i++) {

        c = (a + b) % m;

        a = b;

        b = c;

          if (a == 0 && b == 1) return i + 1;

    }
}

Ossama Ismail - 4 years, 4 months ago

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How many of us have doodled at school adding successive pairs of numbers while keeping only the last digit, wondering if it will never repeat?

1 , 1 , 2 , 3 , 5 , 8 , 3 , 1 , 4 , 5 , 9 , 4 , 3 , 7 , 0 , 7 , 7 , 4 , 1 , 5 , 6 , 1 , 7 , 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7,
8 , 5 , 3 , 8 , 1 , 9 , 0 , 9 , 9 , 8 , 7 , 5 , 2 , 7 , 9 , 6 , 5 , 1 , 6 , 7 , 3 , 0 , 3 , 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3 , 6 , 9 , 5 , 4 , 9 , 3 , 2 , 5 , 7 , 2 , 9 , 1 , 0 , 1 , 1 , 2 , 3 , 5 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5

but it did repeat, and then we wondered why?

Michael Mendrin - 4 years, 4 months ago

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