Poulet Number/Examples/341

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

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $341$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $341$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): pseudoprime
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): pseudoprime