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Zbl 0816.11005
Alford, W.R.; Granville, Andrew; Pomerance, Carl
There are infinitely many Carmichael numbers. (English)
[J] Ann. Math. (2) 139, No.3, 703-722 (1994). ISSN 0003-486X; ISSN 1939-0980/e

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. \par They extend an idea of P. Erd\H{o}s 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 0064.041)]. The question of the existence of products of the form $(*)$ leads to investigations in combinatorial group theory. \par 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)].
[T.Maxsein (Frankfurt / Main)]

MSC 2000:: *11A25 Arithmetic functions, etc.
11N56 Rate of growth of arithmetic functions
11A07 Congruences, etc.
11N69 Distribution of integers in special residue classes

Keywords: divisors of the form $p-1$; Carmichael numbers; existence of infinitely many Carmichael numbers; combinatorial group theory

Citations: Zbl 0064.041; Zbl 0806.11005

Cited in: Zbl 1194.11094 Zbl 1108.11065 Zbl 1085.11047 Zbl 0991.11067 Zbl 0828.11074 Zbl 0816.11006 Zbl 0819.11002

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