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. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $341$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $341$