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Carmichael function
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Carmichael function : ウィキペディア英語版
Carmichael function
In number theory, the Carmichael function of a positive integer ''n'', denoted \lambda(n), is defined as the smallest positive integer ''m'' such that
:a^m \equiv 1 \pmod
for every integer ''a'' that is coprime to ''n''. 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 \psi(n).
The first 36 values of \lambda(n) compared to Euler's totient function \varphi. (in bold if they are different)
==Numerical example==
72 = 49 ≡ 1 (mod 8) because 7 and 8 are coprime (their greatest common divisor equals 1; they have no common factors) and the value of Carmichael's function at 8 is 2. Euler's totient function is 4 at 8 because there are 4 numbers lesser than and coprime to 8 (1, 3, 5, and 7). Whilst it is true that 74 = 2401 ≡ 1 (mod 8), as shown by Euler's theorem, raising 7 to the fourth power is unnecessary because the Carmichael function indicates that 7 squared equals 1 (mod 8). Raising 7 to exponents greater than 2 only repeats the cycle 7, 1, 7, 1, ... . Because the same holds true for 3 and 5, the Carmichael number is 2 rather than 4. 〔http://www25.brinkster.com/denshade/totient.html〕

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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