Definition:Fermat Pseudoprime/Base 5

Definition

Let $q$ be a composite number such that $5^q \equiv 5 \pmod q$.

Then $q$ is a Fermat pseudoprime to base $5$.

Sequence

The sequence of Fermat pseudoprimes base $5$ begins:

$4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, \ldots$

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.