Pisano period: Difference between revisions
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[[Category:Mathematics]] |
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The [[wp:Fibonacci_Number|Fibonacci sequence]] taken modulo 2 is a periodic sequence of period 3 : 0, 1, 1, 0, 1, 1, ... |
The [[wp:Fibonacci_Number|Fibonacci sequence]] taken modulo 2 is a periodic sequence of period 3 : 0, 1, 1, 0, 1, 1, ... |
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Prime numbers are straightforward; the Pisano period of a prime number '''p''' is simply: '''pisano(p)'''. The Pisano period of a composite number '''c''' may be found in different ways. It may be calculated directly: '''pisano(c)''', which works, but may be time consuming to find, especially for larger integers, or, it may be calculated by finding the [[wp:Least common multiple|least common multiple]] of the Pisano periods of each composite component. |
Prime numbers are straightforward; the Pisano period of a prime number '''p''' is simply: '''pisano(p)'''. The Pisano period of a composite number '''c''' may be found in different ways. It may be calculated directly: '''pisano(c)''', which works, but may be time consuming to find, especially for larger integers, or, it may be calculated by finding the [[wp:Least common multiple|least common multiple]] of the Pisano periods of each composite component. |
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;E.G.: |
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'''pisano(m × n)''' is equivalent to '''lcm(pisano(m), pisano(n))''' where '''m''' and '''n''' are '''[[wp:Coprime|coprime]]''' |
'''pisano(m × n)''' is equivalent to '''lcm(pisano(m), pisano(n))''' where '''m''' and '''n''' are '''[[wp:Coprime|coprime]]''' |
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The equation is conjectured, no exceptions have been seen. |
The equation is conjectured, no exceptions have been seen. |
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If a positive integer i is split into its prime factors then the second and first equations above can be applied to generate the pisano period. |
If a positive integer '''i''' is split into its prime factors then the second and first equations above can be applied to generate the pisano period. |
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;Task |
;Task |
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Print pisano(m) for every integer from 1 to 180. |
Print pisano(m) for every integer from 1 to 180. |
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;Related tasks |
;Related tasks |
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* [[Prime decomposition]] |
* [[Prime decomposition]] |
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* [[Least common multiple]] |
* [[Least common multiple]] |
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<br><br> |
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=={{header|Factor}}== |
=={{header|Factor}}== |
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