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# 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.

Synonym:

Knodel number

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11A51*no label found*

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