In number theory, the Carmichael function of a positive integer n, denoted, is defined as the smallest positive integer m such that
for every integer a that is coprime to n. In other words, in more algebraic terms, it defines the exponent of the multiplicative group of integers modulo n. The Carmichael function is also known as the reduced totient function or the least universal exponent function, and is sometimes also denoted .
The first 26 values of (sequence A002322 in OEIS) compared to Euler's totient function :
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 2 | 4 | 2 | 6 | 2 | 6 | 4 | 10 | 2 | 12 | 6 | 4 | 4 | 16 | 6 | 18 | 4 | 6 | 10 | 22 | 2 | 20 | 12 | |
| 1 | 1 | 2 | 2 | 4 | 2 | 6 | 4 | 6 | 4 | 10 | 4 | 12 | 6 | 8 | 8 | 16 | 6 | 18 | 8 | 12 | 10 | 22 | 8 | 20 | 12 |
Read more about Carmichael Function: Numeric Example, Carmichael's Theorem, Hierarchy of Results
Famous quotes containing the words carmichael and/or function:
“An impersonal and scientific knowledge of the structure of our bodies is the surest safeguard against prurient curiosity and lascivious gloating.”
—Marie Carmichael Stopes (1880–1958)
“The mother’s and father’s attitudes toward the child correspond to the child’s own needs.... Mother has the function of making him secure in life, father has the function of teaching him, guiding him to cope with those problems with which the particular society the child has been born into confronts him.”
—Erich Fromm (1900–1980)