Pseudoprime: Difference between revisions

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|<math>V_n(P,Q) - P\ </math> is divisible by <math>n\ </math>
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== Table of smallest Pseudoprimes ==
{| border="1" cellspacing="0"
!colspan="3" align="center"|smallest Pseudoprimes
|-
!Number !!Kind of Pseudoprime !!Bases
|-
|15 ||Fermat pseudoprime ||4, 15
|-
|21 ||Euler pseudoprime ||8, 13
|-
|49 ||strong pseudoprime ||18, 19, 30, 31
|-
|561 ||Carmichael number ||
|-
|1729 ||absolute Euler pseudoprime ||
|}
|}


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A pseudoprime is a composite number that has certain properties in common with prime numbers.

Introduction

If you want to find out if a given number is a prime number, you can to test it based on some properties that all prime numbers share. A property of prime numbers is that they are only divisible by one and itself. This is a defining property: it holds for all prime numbers and no other numbers.

However, other properties hold for all prime numbers and also some other numbers. For instance, every prime number, greater than 3, has the form or (with n an integer), but there are also composite numbers of this form: 25, 35, 49, 55, 65, 77, 85, 91, … . So, you could say that 25, 35, 49, 55, 65, 77, 85, 91, … are pseudoprimes with respect to the property of being of the form or . There exist better properties, which lead to special pseudoprimes:

Different kinds of pseudoprimes

Property kind of pseudoprime
Fermat pseudoprime
Euler pseudoprime
strong pseudoprime
is divisible by Carmichael number
is divisible by Perrin pseudoprime
is divisible by

Table of smallest Pseudoprimes

smallest Pseudoprimes
Number Kind of Pseudoprime Bases
15 Fermat pseudoprime 4, 15
21 Euler pseudoprime 8, 13
49 strong pseudoprime 18, 19, 30, 31
561 Carmichael number
1729 absolute Euler pseudoprime