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There are infinitely many Carmichael numbers. (English) Zbl 0816.11005
Carmichael numbers are those composite integers $n$ for which $a\sp n\equiv a\bmod n$ for every integer $a$. By a result of {\it A. Korselt} [L’intermédiaire des mathématiciens 6, 142--143 (1899)] $n$ is a Carmichael number iff $n$ is squarefree and $p-1$ divides $n-1$ for all primes $p$ dividing $n$. In this paper the authors show the existence of infinitely many Carmichael numbers. They extend an idea of P. Erdős to construct integers $L$ such that $p-1$ divides $L$ for a large number of primes $p$. If there is a product of these primes $\equiv 1\bmod L$, say $$C= p\sb 1\cdot \dots \cdot p\sb k\equiv 1\bmod L \tag $*$ $$ then $C$ is a Carmichael number which is shown by the criterion of A. Korselt mentioned above. In order to find integers with many divisors of the form $p-1$, $p$ prime, the authors generalize a theorem of {\it K. Prachar} [Monatsh. Math. 59, 91--97 (1955; Zbl 0064.04108)]. The question of the existence of products of the form $(*)$ leads to investigations in combinatorial group theory. Reviewer’s remark: For a survey on Carmichael numbers, see the article of {\it C. Pomerance} [Nieuw Arch. Wiskd., IV. Ser. 11, 199--209 (1993; Zbl 0806.11005)].

11A25Arithmetic functions, etc.
11N56Rate of growth of arithmetic functions
11A07Congruences; primitive roots; residue systems
11N69Distribution of integers in special residue classes
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