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 $i^{{m-n}}\equiv 1{\pmod {m}}$. The concept is named after Walter Knödel.^[1]

The set of all n-Knödel numbers is denoted K_n. The special case K₁ represents the Carmichael numbers.

Every composite number m is a n-Knödel number for ${\displaystyle n=m-\varphi (m)}$.

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. Walter Knödel, born May 20th, 1926 in Vienna, earned a Ph.D. in number theory in 1948 (advisors: Edmund Hlawka and Johann Radon) and obtained the habilitation in 1953. Since 1961 he is professor at University of Stuttgart, establishing the new department of computer science. See also The web page on Walter Knödel at the University of Stuttgart.

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.
• Weisstein, Eric W. "Knödel Numbers". MathWorld.