Generalizations of Carmichael numbers I

Abstract

A composite positive integer $n$ is said to be a {\it weak Carmichael number} if $$ \sum_{\gcd(k,n)=1\atop 1\le k\le n-1}k^{n-1}\equiv \varphi(n) \pmod{n}. \leqno(1) $$ It is proved that a composite positive integer $n$ is a weak Carmichael number if and only if $p-1\mid n-1$ for every prime divisor $p$ of $n$. This together with Korselt's criterion yields the fact that every Carmichael number is also a weak Carmichael number. In this paper we mainly investigate arithmetic properties of weak Carmichael numbers. Motivated by the investigations of Carmichael numbers in the last hundred years, here we establish several related results, notions, examples and computatinoal searches for weak Carmichael numbers and numbers closely related to weak Carmichael numbers. Furthermore, using the software {\tt Mathematica 8}, we present the table containing all non-prime powers weak Carmichael numbers less than $2\times 10^6$. Motivated by heuristic arguments, our computations and some old conjectures and results for Carmichael numbers, we propose several conjectures for weak Carmichael numbers and for some other classes of Carmichael like numbers. Finally, we consider weak Carmichael numbers in light of Fermat primality test. We believe that it can be of interest to involve certain particular classes of weak Carmichael numbers in some problems concerning Fermat-like primality tests and the generalized Riemann hypothesis.