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 75479248496430827044831091619765377 81833842440832880856752412600491248324784297704172253450355317535082936750061527 689799541169259849585265122868502865392087298790653950 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.

Title	Poulet number
Canonical name	PouletNumber
Date of creation	2013-03-22 18:11:12
Last modified on	2013-03-22 18:11:12
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	6
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11A51
Synonym	Sarrus number