# 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 Kn.[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

n Kn
1 {561, 1105, 1729, 2465, 2821, 6601, ... } (sequence A002997 in the OEIS)
2 {4, 6, 8, 10, 12, 14, 22, 24, 26, ... } (sequence A050990 in the OEIS)
3 {9, 15, 21, 33, 39, 51, 57, 63, 69, ... } (sequence A033553 in the OEIS)
4 {6, 8, 12, 16, 20, 24, 28, 40, 44, ... } (sequence A050992 in the OEIS)

## References

1. Weisstein, Eric W.. "Knödel Numbers" (in en).

## 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.