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 $n > 2$ . Erdos speculated that this was also true for $n = 1$ but two decades passed before this was conclusively proved by Alford, Granville and Pomerance.

Bibliography

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.