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Carmichael numbers for $$\mathrm{GL}(m)$$. (English) Zbl 07269293
The authors introduce the notation $D_m(n):=\prod_{k=1}^m \Phi_k(n) \quad\text{and}\quad K_m(n):= D_m(n)\times n\prod_{p\mid n} p^{\lceil\log_p m\rceil}$ where $$\Phi_k(X)$$ is the $$k$$th cyclotomic polynomial. It is noted that $$K_m(p)$$ coincides with the exponent of the general linear group $$GL_m(\mathbb{F}_p)$$.
They use these and define an $$m$$-Carmichael number to be a composite $$n$$ for which $$A^{K_m(n)}=I$$ for all $$A\in GL_m(\mathbb{Z}/n\mathbb{Z})$$. This generalizes the ordinary Carmichael numbers, which can now be recognized with $$m=1$$, i.e., $$n$$ is (1-)Carmichael if $$a^{n-1}\equiv 1\pmod n$$ for all $$a$$ relatively prime to $$n$$. With this definition, the analogue of Korselt’s criterion for $$m\geq 2$$ reads “$$n$$ is $$m$$-Carmichael if and only if $$D_m(p)\mid K_m(n)$$ whenever $$p\mid n$$” (Theorem 8).
The infinitude of $$m$$-Carmichael numbers is rather trivial, as any prime power is $$m$$-Carmichael, and so is any power of a given $$m$$-Carmichael number (Proposition 7, Corollary 10). Hence, unlike ordinary Carmichael, $$m$$-Carmichael numbers are not necessarily square-free, nor are they required to be odd – and the even $$m$$-Carmichael numbers are all multiples of 4 (Proposition 11). Families of $$m$$-Carmichael numbers with prescribed prime factors are discussed in Section 3, and all $$m$$-Carmichael numbers less than $$10^5$$ for $$m\in\{2,3,\ldots, 10\}$$ are listed in the Appendix.
Reviewer: Amin Witno (Amman)
##### MSC:
 11A51 Factorization; primality 11Y11 Primality 20G30 Linear algebraic groups over global fields and their integers
##### Keywords:
Carmichael number; pseudoprime; general linear group
OEIS
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##### References:
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