Perrin number: Difference between revisions
imported>Olier Raby (→Properties: Grammar.) |
imported>Karsten Meyer mNo edit summary |
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== Properties == | == Properties == | ||
A special property of the sequence of Perrin numbers is, that if <math>p\ </math> is a prime number, then <math>p\ </math> divides <math>P_p\ </math>. The converse is false, because there exist composite numbers <math>q\ </math> which divide <math>P_q\ </math>. Those numbers <math>q\ </math> are called Perrin pseudoprimes. | A special property of the sequence of Perrin numbers is, that if <math>p\ </math> is a [[prime number]], then <math>p\ </math> divides <math>P_p\ </math>. The converse is false, because there exist composite numbers <math>q\ </math> which divide <math>P_q\ </math>. Those numbers <math>q\ </math> are called Perrin pseudoprimes. | ||
The first few Perrin pseudoprimes are: 271441, 904631, 16532714, 24658561, 27422714, ... | The first few Perrin pseudoprimes are: 271441, 904631, 16532714, 24658561, 27422714, ... | ||
Revision as of 04:32, 19 May 2008
The Perrin numbers are defined by the recurrence relation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_n := \begin{cases} 3 & \mbox{if } n = 0; \\ 0 & \mbox{if } n = 1; \\ 2 & \mbox{if } n = 2; \\ P_{n-2}+P_{n-3} & \mbox{if } n > 2. \\ \end{cases} }
The first few numbers of the sequence are: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, ...
Properties
A special property of the sequence of Perrin numbers is, that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\ } is a prime number, then divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_p\ } . The converse is false, because there exist composite numbers which divide Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_q\ } . Those numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\ } are called Perrin pseudoprimes. The first few Perrin pseudoprimes are: 271441, 904631, 16532714, 24658561, 27422714, ...
