1Math::NumSeq::PisanoPerUisoedr(3C)ontributed Perl DocumeMnattaht:i:oNnumSeq::PisanoPeriod(3)
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NAME

6       Math::NumSeq::PisanoPeriod -- cycle length of Fibonacci numbers mod i
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SYNOPSIS

9        use Math::NumSeq::PisanoPeriod;
10        my $seq = Math::NumSeq::PisanoPeriod->new;
11        my ($i, $value) = $seq->next;
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DESCRIPTION

14       This is the length cycle of Fibonacci numbers modulo i.
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16           1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, ...
17           starting i=1
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19       For example Fibonacci numbers modulo 4 repeat in a cycle of 6 numbers,
20       so value=6.
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22          Fibonacci  0, 1, 1, 2, 3, 5, 8,13,21,34,55,89,144,...
23          mod 4      0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0,...
24                     \--------------/  \--------------/  \---
25                   repeating cycle of 6
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27       The Fibonaccis are determined by a pair F[i],F[i+1] and there can be at
28       most i*i many different pairs mod i, so there's always a finite
29       repeating period.  Since the Fibonaccis can go backwards as
30       F[i-1]=F[i+1]-F[i] the modulo sequence is purely periodic, so the
31       initial 0,1 is always part of the cycle.
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FUNCTIONS

34       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
35       classes.
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37       "$seq = Math::NumSeq::PisanoPeriod->new ()"
38           Create and return a new sequence object.
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40   Random Access
41       "$value = $seq->ith($i)"
42           Return the Pisano period of $i.
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SEE ALSO

45       Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::FibonacciWord
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HOME PAGE

48       <http://user42.tuxfamily.org/math-numseq/index.html>
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LICENSE

51       Copyright 2012, 2013, 2014, 2016 Kevin Ryde
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53       Math-NumSeq is free software; you can redistribute it and/or modify it
54       under the terms of the GNU General Public License as published by the
55       Free Software Foundation; either version 3, or (at your option) any
56       later version.
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58       Math-NumSeq is distributed in the hope that it will be useful, but
59       WITHOUT ANY WARRANTY; without even the implied warranty of
60       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
61       General Public License for more details.
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63       You should have received a copy of the GNU General Public License along
64       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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68perl v5.30.0                      2019-08-05     Math::NumSeq::PisanoPeriod(3)
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