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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)
11A51 Factorization; primality
11Y11 Primality
20G30 Linear algebraic groups over global fields and their integers
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