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Knödel number (Definition)

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$. Erdős 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.



"Knödel number" is owned by PrimeFan. [ full author list (2) | owner history (1) ]
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Other names:  Knodel number

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table of small Knödel numbers $K_n$ for $0 < n < 26$ (Example) by PrimeFan
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Cross-references: Carmichael numbers, coprime, composite, integer, positive
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This is version 3 of Knödel number, born on 2006-07-27, modified 2008-11-29.
Object id is 8181, canonical name is KnodelNumber.
Accessed 1237 times total.

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AMS MSC: 11A51 (Number theory :: Elementary number theory :: Factorization; primality)
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