Lucas-Carmichael number

A Lucas-Carmichael number is a composite, natural number that meets a condition similar to a Carmichael number . It is named after the two mathematicians Édouard Lucas and Robert Daniel Carmichael .

definition

A natural number is called a Lucas-Carmichael number if it fulfills the following properties: ${\displaystyle n\in \mathbb {N} }$

• ${\displaystyle n}$is an odd number
• ${\displaystyle n}$is square-free
• ${\displaystyle n}$has at least 3 prime divisors
• For every prime divisor of the number applies: ${\displaystyle p}$ ${\displaystyle n}$

${\displaystyle p+1}$ Splits ${\displaystyle n+1}$

The number would not be odd and square-free must then would cubes of prime numbers such as or trivial Lucas-Carmichael numbers because for each cube of the three dividers would always divides . ${\displaystyle n}$ ${\displaystyle 2^{3}=8}$ ${\displaystyle 3^{3}=27}$ ${\displaystyle n=p^{3}}$ ${\displaystyle p}$ ${\displaystyle p+1}$ ${\displaystyle n+1=p^{3}+1=(p+1)\cdot (p^{2}-p+1)}$

example

399 = 3 * 7 * 19 and

(3 + 1) = 4 divides 400 = (399 + 1)

(7 + 1) = 8 divides 400 = (399 + 1)

(19 + 1) = 20 divides 400 = (399 + 1)

So 399 is a Lucas-Carmichael number.

The smallest Lucas-Carmichael numbers

The following numbers are Lucas-Carmichael numbers (sequence A006972 in OEIS ):

n Prime divisor 399 = 3 x 7 x 19 935 = 5 x 11 x 17 2015 = 5 * 13 * 31 2915 = 5 x 11 x 53 4991 = 7 * 23 * 31 5719 = 7 * 19 * 43 7055 = 5 * 17 * 83 8855 = 5 x 7 x 11 x 23 12719 = 7 * 23 * 79 18095 = 5 x 7 x 11 x 47

n Prime divisor 20705 = 5 * 41 * 101 20999 = 11 * 23 * 83 22847 = 11 x 31 x 67 29315 = 5 x 11 x 13 x 41 31535 = 5 x 7 x 17 x 53 46079 = 11 x 59 x 71 51359 = 7 x 11 x 23 x 29 60059 = 19 x 29 x 109 63503 = 11 * 23 * 251 67199 = 11 x 41 x 149

n Prime divisor 73535 = 5 x 7 x 11 x 191 76751 = 23 x 47 x 71 80189 = 17 x 53 x 89 81719 = 11 * 17 * 19 * 23 88559 = 19 x 59 x 79 90287 = 17 * 47 * 113 104663 = 13 x 83 x 97 117215 = 5 x 7 x 17 x 197 120581 = 17 x 41 x 173 147455 = 5 x 7 x 11 x 383

n Prime divisor 152279 = 29 * 59 * 89 155819 = 19 * 59 * 139 162687 = 3 x 7 x 61 x 127 191807 = 7 x 11 x 47 x 53 194327 = 7 * 17 * 23 * 71 196559 = 11 107 167 214199 = 23 * 67 * 139 218735 = 5 x 11 x 41 x 97 230159 = 47 x 59 x 83 265895 = 5 x 7 x 71 x 107

n Prime divisor 357599 = 11 * 19 * 29 * 59 388079 = 23 x 47 x 359 390335 = 5 x 11 x 47 x 151 482143 = 31 x 103 x 151 588455 = 5 x 7 x 17 x 23 x 43 653939 = 11 * 13 * 17 * 269 663679 = 31 x 79 x 271 676799 = 19 * 179 * 199 709019 = 17 * 179 * 233 741311 = 53 x 71 x 197

The smallest four prime factor Lucas-Carmichael number is 8,855 = 5 · 7 · 11 · 23.

The smallest five prime factor Lucas-Carmichael number is 588,455 = 5 7 17 23 43.

The smallest Lucas-Carmichael number with six prime factors is 139,501,439 = 7 11 17 19 71 79.

The smallest Lucas-Carmichael number with seven prime factors is 3,512,071,871 = 7 11 17 23 31 53 71.

The smallest Lucas-Carmichael number with eight prime factors is 199,195,047,359 = 7 11 17 19 23 31 47 239.

properties

• Because of the identity, the following applies for every prime divisor of a natural number : ${\displaystyle n+1=-{\frac {n}{p}}+1+(p+1){\frac {n}{p}}}$ ${\displaystyle p}$ ${\displaystyle n}$

${\displaystyle n+1\equiv -{\frac {n}{p}}+1\ mod\ p+1}$.

Thus an odd square-free number is a Lucas-Carmichael number if and only if for each of its prime divisors: divides . ${\displaystyle n}$ ${\displaystyle p+1}$ ${\displaystyle {\frac {n}{p}}-1}$

• There are Fermat pseudoprimes among the Lucas-Carmichael numbers.
• Lucas-Carmichael numbers are not a subset of the Fermat pseudoprime numbers .
• It is not known whether a Lucas-Carmichael number that is also a Carmichael number exists.

Web links

• Lucas-Carmichael number . In: PlanetMath . (English)