Carmichael numbers are those composite integers for which for every integer . By a result of A. Korselt [L’intermédiaire des mathématiciens 6, 142-143 (1899)] is a Carmichael number iff is squarefree and divides for all primes dividing . In this paper the authors show the existence of infinitely many Carmichael numbers.
They extend an idea of P. Erdős to construct integers such that divides for a large number of primes . If there is a product of these primes , say
then 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 , 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 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)].