# Fermat Pseudoprime/Base 3/Examples/91

## Theorem

The smallest Fermat pseudoprime to base $3$ is $91$:

$3^{91} \equiv 3 \pmod {91}$

despite the fact that $91$ is not prime:

$91 = 7 \times 13$

## Proof

We have that:

 $\displaystyle 3^{91}$ $=$ $\displaystyle 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 347$ $\displaystyle$ $=$ $\displaystyle 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 344 + 3$ $\displaystyle$ $=$ $\displaystyle 91 \times 287 \, 735 \, 062 \, 684 \, 210 \, 299 \, 754 \, 661 \, 455 \, 367 \, 364 \, 637 \, 583 \, 784 + 3$ $\displaystyle \leadsto \ \$ $\displaystyle 3^{91}$ $\equiv$ $\displaystyle 3 \pmod {91}$