Cullen number: Difference between revisions
(Created page with '{{natural number property}} ==Definition== Suppose <math>n</math> is a natural number. The '''Cullen number''' parametrized by <math>n</math>, denoted <math>C_n</math>, is …') |
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<math>C_n := n \cdot 2^n + 1</math> | <math>C_n := n \cdot 2^n + 1</math> | ||
A Cullen number that is a prime is termed a [[Cullen prime]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Stronger than::Proth number]]: A Proth number is a number of the form <math>k \cdot 2^n + 1</math>, <math>k < 2^n</math>. Since <math>n < 2^n</math> for all <math>n</math>, any Cullen number is a Proth number. | * [[Stronger than::Proth number]]: A Proth number is a number of the form <math>k \cdot 2^n + 1</math>, <math>k < 2^n</math>. Since <math>n < 2^n</math> for all <math>n</math>, any Cullen number is a Proth number. | ||
* [[Stronger than::Generalized Cullen number]]: A Cullen number of the form <math>n \cdot b^n + 1</math> where <math>n + 2 > b</math>. Note that the special case where <math>b = 2</math> gives Cullen numbers. | |||
===Other related properties=== | ===Other related properties=== | ||
* [[Fermat number]] is a number of the form <math>2^{2^n} + 1</math>, and is also a special case of a Proth number. | * [[Fermat number]] is a number of the form <math>2^{2^n} + 1</math>, and is also a special case of a Proth number. | ||
* [[Sierpinski number]] is a number of the form <math>k \cdot 2^n - 1</math>, with <math>k < 2^n</math>. | * [[Sierpinski number]] is a number of the form <math>k \cdot 2^n - 1</math>, with <math>k < 2^n</math>. | ||
Latest revision as of 00:01, 30 May 2010
This article defines a property that can be evaluated for a natural number, i.e., every natural number either satisfies the property or does not satisfy the property.
View a complete list of properties of natural numbers
Definition
Suppose is a natural number. The Cullen number parametrized by , denoted , is defined as:
A Cullen number that is a prime is termed a Cullen prime.
Relation with other properties
Weaker properties
- Proth number: A Proth number is a number of the form , . Since for all , any Cullen number is a Proth number.
- Generalized Cullen number: A Cullen number of the form where . Note that the special case where gives Cullen numbers.
- Fermat number is a number of the form , and is also a special case of a Proth number.
- Sierpinski number is a number of the form , with .
