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Curious Properties of an Iterative Process

Ryohei Miyadera
Organization: Kwansei Gakuin High School
Department: Mathematics Department
Daisuke Minematsu
Organization: Osaka University
Satoshi Hashiba
Munetoshi Sakaguchi
Kwansei Gakuin
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We define a function f(n)=(n1)^(n1)+(n2)^(n2)+..., (nk)^(nk), where{ n1, n2, ..., nk } is the list of the digits of a natural number n. If we start with a natural number n, and repeatedly apply the function f, then we can generate a sequence {n, f(n), f(f(n)), f(f(f(n))), }.

For any natural number n the sequence {n, f(n), f(f(n)), f(f(f(n))), } eventually enters into one of 8 loops. The fact that there are only 8 loops including 2 fixed points is very curious. This fact was found by 7 high school students, and was proved using mathematics and calculation by Mathematica.

*Mathematics > Number Theory
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*Curious properties of an iterative process under various moduli   [in MathSource: Packages and Programs]
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Iterative Process.nb (277.9 KB) - Mathematica Notebook