Poulet Number

A Fermat Pseudoprime to base 2, denoted psp(2), i.e., a Composite Odd Integer such that

$$2^{n-1}\equiv 1\ \left({{\rm mod\ } {n}}\right).$$

The first few Poulet numbers are 341, 561, 645, 1105, 1387, ... (Sloane's A001567). Pomerance et al. (1980) computed all 21,853 Poulet numbers less than $25\times 10^9$.

Pomerance has shown that the number of Poulet numbers less than $x$ for sufficiently large $x$ satisfy

$$\mathop{\rm exp}\nolimits [(\ln x)^{5/14}]<P_2(x)<x \mathop{... ...xp}\nolimits \left({-{\ln x\ln\ln\ln x\over 2\ln\ln x}}\right)$$

(Guy 1994).

A Poulet number all of whose Divisors $d$ satisfy $d\vert 2^d-2$ is called a Super-Poulet Number. There are an infinite number of Poulet numbers which are not Super-Poulet Numbers. Shanks (1993) calls any integer satisfying $2^{n-1}\equiv 1\ \left({{\rm mod\ } {n}}\right)$ (i.e., not limited to Odd composite numbers) a Fermatian.

See also Fermat Pseudoprime, Pseudoprime, Super-Poulet Number

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 28-29, 1994.

Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. ``The Pseudoprimes to $25\cdot 10^9$.'' Math. Comput. 35, 1003-1026, 1980. Available electronically from ftp://sable.ox.ac.uk/pub/math/primes/ps2.Z.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 115-117, 1993.

Sloane, N. J. A. Sequence A001567/M5441 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.