Poulet Number/Examples/341

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Theorem

The smallest Poulet number is $341$:

$2^{341} \equiv 2 \pmod {341}$

despite the fact that $341$ is not prime:

$341 = 11 \times 31$


Proof

We have that:

\(\ds 2^{341}\) \(=\) \(\ds 4 \, 479 \, 489 \, 484 \, 355 \, 608 \, 421 \, 114 \, 884 \, 561 \, 136 \, 888 \, 556 \, 243 \, 290 \, 994 \, 469 \, 299 \, 069 \, 799 \, 978 \, 201 \, 927 \, 583 \, 742 \, 360 \, 321 \, 890 \, 761 \, 754 \, 986 \, 543 \, 214 \, 231 \, 552\)
\(\ds \) \(=\) \(\ds 4 \, 479 \, 489 \, 484 \, 355 \, 608 \, 421 \, 114 \, 884 \, 561 \, 136 \, 888 \, 556 \, 243 \, 290 \, 994 \, 469 \, 299 \, 069 \, 799 \, 978 \, 201 \, 927 \, 583 \, 742 \, 360 \, 321 \, 890 \, 761 \, 754 \, 986 \, 543 \, 214 \, 231 \, 550 + 2\)
\(\ds \) \(=\) \(\ds 341 \times 13 \, 136 \, 332 \, 798 \, 696 \, 798 \, 888 \, 899 \, 954 \, 724 \, 741 \, 608 \, 669 \, 335 \, 164 \, 206 \, 654 \, 835 \, 981 \, 818 \, 117 \, 894 \, 215 \, 788 \, 100 \, 763 \, 407 \, 304 \, 286 \, 671 \, 514 \, 789 \, 484 \, 550 + 2\)
\(\ds \leadsto \ \ \) \(\ds 2^{341}\) \(\equiv\) \(\ds 2 \pmod {341}\)


This needs considerable tedious hard slog to complete it.
In particular: It remains to be shown that it is the smallest.
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Sources

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