Poulet number

A Poulet number or Sarrus number is a composite integer $n$ such that $2^{n}\\equiv 2\\mod n$ . In other words, a base 2 pseudoprime (thus a Poulet number that satisfies the congruence for other bases is a Carmichael number). The first few Poulet numbers are 341, 561, 645, 1105, 1387, 1729, 1905, listed in A001567 of Sloane’s OEIS.

For example, 561 is a Poulet number, since $2^{561}-2$ is 7547924849643082704483109161976537781833842440832880856752412600491248324784297704172253450355317535082936750061527689799541169259849585265122868502865392087298790653950 and that’s divisible by 561. The number 561 is not prime, it has the prime factors 3, 11, and 17.

Poulet numbers are counterexamples to the Chinese hypothesis.

References

1 Derrick Henry Lehmer, “Errata for Poulet’s table,” Math. Comp. 25 25 (1971): 944 - 945.

单词	PouletNumber
释义	Poulet number A Poulet number or Sarrus number is a composite integer $n$ such that $2^{n}\\equiv 2\\mod n$ . In other words, a base 2 pseudoprime (thus a Poulet number that satisfies the congruence for other bases is a Carmichael number). The first few Poulet numbers are 341, 561, 645, 1105, 1387, 1729, 1905, listed in A001567 of Sloane’s OEIS. For example, 561 is a Poulet number, since $2^{561}-2$ is 7547924849643082704483109161976537781833842440832880856752412600491248324784297704172253450355317535082936750061527689799541169259849585265122868502865392087298790653950 and that’s divisible by 561. The number 561 is not prime, it has the prime factors 3, 11, and 17. Poulet numbers are counterexamples to the Chinese hypothesis. References 1 Derrick Henry Lehmer, “Errata for Poulet’s table,” Math. Comp. 25 25 (1971): 944 - 945.
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