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Knödel Numbers

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For every k>=1, let C_k be the set of composite numbers n>k such that if 1<a<n, GCD(a,n)=1 (where GCD is the greatest common divisor), then a^(n-k)=1 (mod n).

Special cases include C_1, which is the set of Carmichael numbers, and C_3, which gives the D-numbers.

Makowski (1962/1963) proved that there are infinitely many members of C_k for k>=2. The following table summarized Knödel numbers C_k for small k.

kOEISC_k
1A002997561, 1105, 1729, 2465, 2821, 6601, 8911, ...
2A0509904, 6, 8, 10, 12, 14, 22, 24, 26, 30, ...
3A0335539, 15, 21, 33, 39, 51, 57, 63, 69, 87, ...
4A0509926, 8, 12, 16, 20, 24, 28, 40, 44, 48, ...
5A05099325, 65, 85, 145, 165, 185, 205, ...

See also

Carmichael Number, D-Number, Greatest Common Divisor

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References

Makowski, A. "Generalization of Morrow's D-Numbers." Simon Stevin 36, 71, 1962/1963.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 125-126, 1989.Sloane, N. J. A. Sequences A002997/M5462, A033553, A050990, A050992, and A050993 in "The On-Line Encyclopedia of Integer Sequences."

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Knödel Numbers

Cite this as:

Weisstein, Eric W. "Knödel Numbers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KnoedelNumbers.html

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