Knödel number

In number theory, an n-Knödel number for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies ${\displaystyle i^{m-n}\equiv 1{\pmod {m}}}$.^[1] The concept is named after Walter Knödel.^{[citation needed]}

The set of all n-Knödel numbers is denoted K_n.^[1]The special case K₁ is the Carmichael numbers.^[1] There are infinitely many n-Knödel numbers for a given n.

Due to Euler's theorem every composite number m is an n-Knödel number for ${\displaystyle n=m-\varphi (m)}$ where ${\displaystyle \varphi }$ is Euler's totient function.

Examples

References

1. Weisstein, Eric W. "Knödel Numbers". mathworld.wolfram.com. Retrieved 2021-09-14.

Literature

• Makowski, A (1963). Generalization of Morrow's D-Numbers. p. 71.
• Ribenboim, Paulo (1989). The New Book of Prime Number Records. New York: Springer-Verlag. p. 101. ISBN 978-0-387-94457-9.