Generalizations of Carmichael numbers I

TL;DR

This paper introduces weak Carmichael numbers, explores their properties, provides computational data, and discusses their implications for primality testing and related conjectures.

Contribution

It characterizes weak Carmichael numbers, establishes their relation to Carmichael numbers, and presents computational results and conjectures about their properties.

Findings

Weak Carmichael numbers satisfy p-1 | n-1 for all prime divisors p.

All Carmichael numbers are also weak Carmichael numbers.

Computationally identified all non-prime power weak Carmichael numbers less than 2 million.

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.