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| In [[number theory]], a '''Knödel number''' for a given [[positive integer]] ''n'' is a [[composite number]] ''m'' with the property that each ''i'' < ''m'' [[coprime]] to ''m'' satisfies <math>i^{m - n} \equiv 1 \pmod{m}</math>. The concept is named after Walter Knödel.<ref>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 [http://www.uni-stuttgart.de/hkom/publikationen/uni-kurier/uk98/leute/lt102b.html The web page on Walter Knödel] at the [[University of Stuttgart]].</ref> The [[Set (mathematics)|set]] of all Knödel numbers for ''n'' is denoted ''K''<sub>''n''</sub>.
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| The special case ''K''<sub>1</sub> are the [[Carmichael number]]s.
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| == Examples ==
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| {|class="wikitable"
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| !''n'' !! colspan="2" | ''K''<sub>''n''</sub>
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| |1 ||{561, 1105, 1729, 2465, 2821, 6601, ... }|| {{OEIS|id=A002997}}
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| |2 ||{4, 6, 8, 10, 12, 14, 22, 24, 26, ... }|| {{OEIS|id=A050990}}
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| |3 ||{9, 15, 21, 33, 39, 51, 57, 63, 69, ... }|| {{OEIS|id=A033553}}
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| |4 ||{6, 8, 12, 16, 20, 24, 28, 40, 44, ... }||{{OEIS|id= A050992}}
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| |}
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| == Literature ==
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| * {{cite book |title=Generalization of Morrow's D-Numbers |last=Makowski |first=A |year=1963 |page=71}}
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| * {{cite book |title=The New Book of Prime Number Records |last=Ribenboim |first=Paulo |authorlink=Paulo Ribenboim |year=1989 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-94457-9 |page=101}}
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| * {{mathworld|title=Knödel Numbers|urlname=KnoedelNumbers}}
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| == References ==
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| <references/>
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| {{Classes of natural numbers}}
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| {{DEFAULTSORT:Knodel Number}}
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| [[Category:Number theory]]
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| {{numtheory-stub}}
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