Definition:Fermat Pseudoprime/Base 3
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Definition
Let $q$ be a composite number such that $3^q \equiv 3 \pmod q$.
Then $q$ is a Fermat pseudoprime to base $3$.
Sequence
The sequence of Fermat pseudoprimes base $3$ begins:
- $91, 121, 286, 671, 703, \ldots$
Examples
$91$ is a Fermat Pseudoprime to Base $3$
The smallest Fermat pseudoprime to base $3$ is $91$:
- $3^{91} \equiv 3 \pmod {91}$
despite the fact that $91$ is not prime:
- $91 = 7 \times 13$
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.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $91$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $91$