Knödel number

The Knödel numbers $K_{n}$ for a given positive integer $n$ are the set of composite integers $m>n$ such that any $b<m$ coprime to $m$ satisfies $b^{m-n}\equiv 1\mod m$ . The Carmichael numbers are $K_{1}$ . There are infinitely many Knodel number $K_{n}$ for a given $n$ , something which was first proven only for $n>2$ . Erdős speculated that this was also true for $n=1$ but two decades passed before this was conclusively proved by Alford, Granville and Pomerance.

References

1 W. R. Alford, A. Granville, and C. Pomerance. “There are Infinitely Many Carmichael Numbers” Annals of Mathematics 139 (1994): 703 - 722
2 P. Ribenboim, The Little Book of Bigger Primes, (2004), New York: Springer-Verlag, p. 102.

Title	Knödel number
Canonical name	KnodelNumber
Date of creation	2013-03-22 16:06:54
Last modified on	2013-03-22 16:06:54
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	6
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A51
Synonym	Knodel number