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On the number of prime divisors of higher-order Carmichael numbers. (English) Zbl 1096.11003
The authors give lower bounds for the number of prime divisors of higher order Carmichael numbers as defined by E. W. Howe [“Higher-order Carmichael numbers”, Math. Comput. 69, 1711–1719 (2000; Zbl 0966.11006)]. A composite number $$n$$ is a Carmichael number of order $$m$$ if and only if $$n$$ is squarefree and for every prime divisor $$p$$ of $$n$$ and for every integer $$r$$ with $$1\leq r\leq m$$, there is an integer $$i_r\geq 0$$ such that $$n\equiv p^{i_r}\pmod{p^r-1}$$. It is a rigid Carmichael number of order $$m$$ if all $$i_r=0$$ in this characterization. The authors prove that every Carmichael number of order $$m$$ has at least $$m+2$$ different prime divisors. Furthermore, they prove that if $$n$$ is a Carmichael number of order $$m\geq 2$$ and the largest prime divisor of $$n$$ is larger than some given constant depending on $$m$$, then $$n$$ has at least $$s/2$$ different prime divisors, where $$s=\sum_{k\leq m}\varphi(k)$$ and $$\varphi$$ denotes the Euler totient function. Finally, a rigid Carmichael number of order $$m\geq 2$$ has at least $$s+1$$ different prime divisors.
##### MSC:
 11A51 Factorization; primality 11A25 Arithmetic functions; related numbers; inversion formulas
##### Keywords:
Carmichael number