# 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 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 KnodelNumber 2013-03-22 16:06:54 2013-03-22 16:06:54 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 11A51 Knodel number