Given a positive integer , the composite integers such that any coprime to satisfies form the infinite set of Knödel numbers . The Carmichael numbers are . There are infinitely many Knodel number for a given , something which was first proven only for . Erdős speculated that this was also true for 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.