In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Lucas–Carmichael number is a positive
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic material ...
integer ''n'' such that
# If ''p'' is a
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of ''n'', then ''p'' + 1 is a factor of ''n'' + 1;
# ''n'' is odd and
square-free {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
...
.
The first condition resembles Korselt's criterion for
Carmichael number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
: b^n\equiv b\pmod
for all integers . The relation may also be expressed in the form:
: b^\equiv 1\pmod
for all integers b ...
s, 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
__NOTOC__
François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.
Biography
Luc ...
and
Robert Carmichael
Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was an American mathematician.
Biography
Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was s ...
.
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
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
: b^n\equiv b\pmod
for all integers . The relation may also be expressed in the form:
: b^\equiv 1\pmod
for all integers b ...
.
Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers.
If we let
denote the number of Lucas–Carmichael numbers up to
, Wright showed that there exists a positive constant
such that
.
List of Lucas–Carmichael numbers
The first few Lucas–Carmichael numbers and their prime factors are listed below.
References
External links
*
*
*
{{DEFAULTSORT:Lucas-Carmichael number
Eponymous numbers in mathematics
Integer sequences