Editing Lucas–Carmichael number (section)

== Properties ==
The smallest Lucas–Carmichael number is 399 = 3 × 7 × 19. It is easy to verify that 3+1, 7+1, and 19+1 are all factors of 399+1 = 400.

The smallest Lucas–Carmichael number with 4 factors is 8855 = 5 × 7 × 11 × 23.

The smallest Lucas–Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43.

It is not known whether any Lucas–Carmichael number is also a [[Carmichael number]].

[[Thomas Wright (mathematician)|Thomas Wright]] proved in 2016 that there are infinitely many Lucas–Carmichael numbers.<ref>{{cite journal |author=Thomas Wright |title=There are infinitely many elliptic Carmichael numbers |journal=[[Bull. London Math. Soc.]] |volume=50 |year=2018 |issue=5 |pages=791–800 |arxiv=1609.00231 |doi=10.1112/blms.12185 |s2cid=119676706 }}</ref>  If we let <math> N(X)</math> denote the number of Lucas–Carmichael numbers up to <math> X</math>, Wright showed that there exists a positive constant <math>K</math> such that

<math> N(X) \gg X^{K/\left( \log\log \log X\right)^2}</math>.