Editing Arithmetic function (section)

=== ''λ''(''n'') – Carmichael function ===
'''[[Carmichael function|''λ''(''n'')]]''', the Carmichael function, is the smallest positive number such that <math>a^{\lambda(n)}\equiv 1 \pmod{n}</math> &nbsp; for all ''a'' coprime to ''n''. Equivalently, it is the [[least common multiple]] of the orders of the elements of the [[Multiplicative group of integers modulo n|multiplicative group of integers modulo ''n'']].

For powers of odd primes and for 2 and 4, ''λ''(''n'') is equal to the Euler totient function of ''n''; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of ''n'':
<math display="block">\lambda(n) = \begin{cases}
\;\;\phi(n) &\text{if }n = 2,3,4,5,7,9,11,13,17,19,23,25,27,\dots\\
\tfrac 1 2 \phi(n)&\text{if }n=8,16,32,64,\dots
\end{cases}</math>
and for general ''n'' it is the least common multiple of ''λ'' of each of the prime power factors of ''n'':
<math display="block">\lambda(p_1^{a_1}p_2^{a_2} \dots p_{\omega(n)}^{a_{\omega(n)}}) = \operatorname{lcm}[\lambda(p_1^{a_1}),\;\lambda(p_2^{a_2}),\dots,\lambda(p_{\omega(n)}^{a_{\omega(n)}}) ].</math>