Pseudoprime

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

Introduction

To find out if a given number is a prime number, it can be tested for properties that all prime numbers share. One property of a prime number is that it is 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, we can 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, as follows.

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, 11
21 Euler pseudoprime 8, 13
49 strong pseudoprime 18, 19, 30, 31
561 Carmichael number
1729 absolute Euler pseudoprime