Carmichael function: Difference between revisions

Revision as of 18:16, 16 April 2022

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===Exponential cycle length===

If <math>r_{\mathrm{mvar | nmax}}=\max_i\{r_i\}</math> hasis maximumthe primebiggest exponent in the prime factorization <math> n= p_1^{r_1}p_2^{mathr_2} |\cdots p_{k}^{r_k} ''r''<sub>max</submath>}} underof prime{{mvar factorization| n}}, then for all {{mvar | a}} (including those not coprime to {{mvar | n}}) and all {{math | ''r'' ≥ ''r''_max}},

:<math>a^r \equiv a^{r+\lambda(n)+r} \pmod n.</math>

In particular, for [[Square-free integer|square-free]] {{mvar | n}} ({{math | ''r''_max {{=}} 1}}), for all {{mvar | a}} we have