# Definition:Poulet Number

## Contents

## Definition

Let $q$ be a composite number such that $2^q \equiv 2 \pmod q$.

Then $q$ is a **Poulet number**.

### Sequence of Poulet Numbers

The sequence of Poulet numbers begins:

- $341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, \ldots$

## Examples of Poulet Numbers

### $341$ is a Poulet Number

The smallest Poulet number is $341$:

- $2^{341} \equiv 2 \pmod {341}$

despite the fact that $341$ is not prime:

- $341 = 11 \times 31$

## Also known as

**Poulet numbers** are also referred to as **pseudoprimes** or **Fermat pseudoprimes** to base $2$.

Some sources refer to them as **Sarrus numbers**, for Pierre Frédéric Sarrus.

## Source of Name

This entry was named for Paul Poulet.

## Historical Note

From as far back as the ancient Chinese, right up until the time of Gottfried Wilhelm von Leibniz, it was thought that $n$ had to be prime in order for $2^n - 2$ to be divisible by $n$.

This used to be used as a test for primality.

But it was discovered that $2^{341} \equiv 2 \pmod {341}$, and $341 = 31 \times 11$ and so is composite.

The first person to point this out was Pierre Frédéric Sarrus, in whose honour these numbers are sometimes called **Sarrus numbers**.

Paul Poulet set about calculating these **Fermat pseudoprimes to base $2$**, first up to $50$ million in $1926$, then up to $100$ million in $1938$.

They have since been named **Poulet numbers** in honour of that feat of calculation.

It has been determined by John Lewis Selfridge and Samuel Standfield Wagstaff Jr. that in the first $20 \, 000 \, 000 \, 000$ positive integers there are no more than $19 \, 865$ Poulet numbers.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $341$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $341$

- Weisstein, Eric W. "Poulet Number." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/PouletNumber.html