zbMATH — the first resource for mathematics

Carmichael numbers for \(\mathrm{GL}(m)\). (English) Zbl 1464.11011
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 \(A\in \mathrm{GL}_m(\mathbb{Z}/n\mathbb{Z})\). 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
Full Text: Link
[1] W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers,Ann. of Math. (2)139(1994), 703-722. · Zbl 0816.11005
[2] Y. Ge, Elementary properties of cyclotomic polynomials,Math. Reflec.2(2008). 9
[3] E. W. Howe, Higher-order Carmichael numbers,Math. Comp.69(2000), 1711-1719. · Zbl 0966.11006
[4] J. B. Marshall, On the extension of Fermat’s theorem to matrices of ordern,Proc. Edinb. Math. Soc.6(1939), 85-91. · Zbl 0022.11203
[5] R. J. McIntosh, M. Dipra, Carmichael numbers withp+ 1|n+ 1,J. Number Theory 147(2015), 81-91. · Zbl 1303.11017
[6] I. Niven, Fermat’s theorem for matrices,Duke Math. J.15(1948), 823-826. · Zbl 0032.00102
[7] G. A. Steele, Carmichael numbers in number rings,J. Number Theory128(2008), 910-917. · Zbl 1176.11049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.