Lucas–Carmichael number

In mathematics, a Lucas–Carmichael number is a positive composite integer ''n'' such that
# if ''p'' is a prime factor of ''n'', then ''p'' + 1 is a factor of ''n'' + 1;
# ''n'' is odd and square-free.
The first condition resembles the Korselt's criterion for Carmichael numbers, where -1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since ''n''³ + 1 = (''n'' + 1)(''n''² − ''n'' + 1) is always divisible by ''n'' + 1).

They are named after Édouard Lucas and Robert Carmichael.

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 proved in 2016 that there are infinitely many Lucas–Carmichael numbers. If we let $N(X)$ denote the number of Lucas–Carmichael numbers up to $X$, Wright showed that there exists a positive constant $K$ such that

$N(X) \gg X^$.

List of Lucas–Carmichael numbers

The first few Lucas–Carmichael numbers and their prime factors are listed below.

References

External links

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{{DEFAULTSORT:Lucas-Carmichael number
Integer sequences