# Fermat Pseudoprime/Base 3/Examples/91

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## 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:

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

Here we define Fermat pseudoprime to base $3$ to be a composite number $n$ such that (a slightly stronger result):

$3^{n - 1} \equiv 1 \pmod n$

Otherwise we have:

$3^6 = 729 \equiv 3 \pmod 6$

Following our definition, we see that a Fermat pseudoprime to base $3$ cannot be divisible by $3$:

 $\ds 3^{3 k - 1}$ $\equiv$ $\ds 0$ $\ds \pmod 3$ $\ds$ $\not \equiv$ $\ds 1$ $\ds \pmod {3 k}$

neither can it be divisible by $4$:

 $\ds 3^{4 k}$ $=$ $\ds \paren {3^{2 k} }^2$ $\ds$ $\equiv$ $\ds 1$ $\ds \pmod 4$ Square Modulo 4 $\ds$ $\not \equiv$ $\ds 3$ $\ds \pmod {4 k}$

nor $10$:

 $\ds 3^{10 k}$ $=$ $\ds \paren {3^{5 k} }^2$ $\ds$ $\equiv$ $\ds \pm 1$ $\ds \pmod 5$ Square Modulo 5 $\ds$ $\not \equiv$ $\ds 3$ $\ds \pmod {10 k}$

Also such a number cannot be equal to twice a prime greater than $3$:

 $\ds 3^{2 p}$ $\equiv$ $\ds 3^2$ $\ds \pmod p$ Fermat's Little Theorem $\ds$ $\not \equiv$ $\ds 3$ $\ds \pmod p$ Since $p \nmid 6 = 2 \times 3$ $\ds \leadsto \ \$ $\ds 3^{2 p}$ $\not \equiv$ $\ds 3$ $\ds \pmod {2 p}$

nor five times a prime greater than $5$:

 $\ds 3^{5 p}$ $\equiv$ $\ds 3^5$ $\ds \pmod p$ Fermat's Little Theorem $\ds$ $\not \equiv$ $\ds 3$ $\ds \pmod p$ Since $p \nmid 240 = 2^4 \times 3 \times 5$ $\ds \leadsto \ \$ $\ds 3^{5 p}$ $\not \equiv$ $\ds 3$ $\ds \pmod {5 p}$

Therefore only the following numbers less than $91$ remain:

$25, 49, 77$

and we check:

 $\ds 3^{25}$ $=$ $\ds 847 \, 288 \, 609 \, 443$ $\ds$ $=$ $\ds 33 \, 891 \, 544 \, 377 \times 25 + 18$ $\ds \leadsto \ \$ $\ds 3^{25}$ $\equiv$ $\ds 18 \pmod {25}$ $\ds 3^{49}$ $=$ $\ds 239 \, 299 \, 329 \, 230 \, 617 \, 529 \, 590 \, 083$ $\ds$ $=$ $\ds 4 \, 883 \, 659 \, 780 \, 216 \, 684 \, 277 \, 348 \times 49 + 31$ $\ds \leadsto \ \$ $\ds 3^{49}$ $\equiv$ $\ds 31 \pmod {49}$ $\ds 3^{77}$ $=$ $\ds 5 \, 474 \, 401 \, 089 \, 420 \, 219 \, 382 \, 077 \, 155 \, 933 \, 569 \, 751 \, 763$ $\ds$ $=$ $\ds 71 \, 096 \, 118 \, 044 \, 418 \, 433 \, 533 \, 469 \, 557 \, 578 \, 827 \, 944 \times 77 + 75$ $\ds \leadsto \ \$ $\ds 3^{77}$ $\equiv$ $\ds 75 \pmod {77}$

hence the result.

$\blacksquare$