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There are infinitely many Carmichael numbers. (English) Zbl 0816.11005

Carmichael numbers are those composite integers $n$ for which ${a}^{n}\equiv a\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}n$ 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 $\equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}L$, say

$C={p}_{1}·\cdots ·{p}_{k}\equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}L\phantom{\rule{2.em}{0ex}}\left(*\right)$

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 0064.041)]. The question of the existence of products of the form $\left(*\right)$ 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, etc. 11N56 Rate of growth of arithmetic functions 11A07 Congruences; primitive roots; residue systems 11N69 Distribution of integers in special residue classes