Carmichael function: Difference between revisions

Revision as of 06:52, 16 April 2022

Revision as of 08:09, 16 April 2022

Line 65:

:<math> n= p_1^{r_1}p_2^{r_2} \cdots p_{k}^{r_k} </math>

where {{math | ''p''₁ < ''p''₂ < ... < ''p_k''}} are [[prime]]s and {{math | ''r''₁, ''r''₂, ..., ''r_k''}} are positive integers. Then {{math | ''λ''(''n'')}} is the [[least common multiple]] of the {{mvar | λ}} of each of its prime power factors:

:<math>\lambda(n) = \operatorname{lcm}\leftBigl(\lambda\left(p_1^{r_1}\right),\lambda\left(p_2^{r_2}\right),\ldots,\lambda\left(p_k^{r_k}\right)\rightBigr).</math>

This can be proved using the [[Chinese remainder theorem]].

Line 71:

:<math>\lambda(p^r) = \begin{cases}

\tfrac12\varphi\left(p^r\right)&\text{if }p=2,\land r\geq 3 \;(\mbox{i.e. }(p^r = 8,16,32,64,128,256,\dots)\\

\varphi\left(p^r\right) &\mbox{otherwise }\;(\mbox{i.e. }p^r = 2,4,3^r,4,5^r,7^r,11^r,13^r,17^r,19^r,23^r,29^r,31^r,\dots)

\end{cases}</math>