# zbMATH — the first resource for mathematics

There are infinitely many Carmichael numbers. (English) Zbl 0816.11005
Carmichael numbers are those composite integers $$n$$ for which $$a^ n\equiv a\bmod n$$ for every integer $$a$$. By a result of 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 $C= p_1\cdots p_k\equiv 1\bmod L \tag {*}$ 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 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 C. Pomerance [Nieuw Arch. Wiskd., IV. Ser. 11, 199–209 (1993; Zbl 0806.11005)].

##### MSC:
 11A25 Arithmetic functions; related numbers; inversion formulas 11N56 Rate of growth of arithmetic functions 11A07 Congruences; primitive roots; residue systems 11N69 Distribution of integers in special residue classes
Full Text: