91 is Pseudoprime to 35 Bases less than 91
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Theorem
$91$ is a Fermat pseudoprime in $35$ bases less than itself:
- $3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$
Proof
By definition of a Fermat pseudoprime, we need to check for $a < 91$:
- $a^{90} \equiv 1 \pmod {91}$
is satisfied or not.
Note that:
\(\ds \paren {91 - a}^{90}\) | \(=\) | \(\ds {-a}^{90}\) | Congruence of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds a^{90}\) |
So we essentially need to check $2 \le a \le 45$ only.
Since $91 = 7 \times 13$, we can also exclude all multiples of $7$ and $13$.
Historical Note
This result is attributed by David Wells in his $1997$ work Curious and Interesting Numbers, 2nd ed. to Tiger Redman, but no corroboration can be found for this on the internet.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $91$