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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 amodn 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 1modL, say

C=p 1 ··p k 1modL(*)

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