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Curious properties of an iterative process under various moduli

Ryohei Miyadera
Organization: Kwansei Gakuin High School
Department: Mathematics Department
Satoshi Hashiba
Revision date


Let K be an arbitrary positive number. We define a function

f(n) = Mod[(n1)^(n1) + (n2)^(n2) + ..., (nk)^(nk),K],

where { n1,n2, ..., nk } is the list of the digits of a natural number n. If we repeatedly apply the function f, then we can generate a sequence {n, f(n), f(f(n)), f(f(f(n))), .....}.

If we choose a proper number K, then for any natural number n the sequence {n, f(n), f(f(n)), f(f(f(n))), ...} eventually converges to 1.

This curious fact is thoroughly explored in Mathematica.

*Mathematics > Number Theory

iterative process
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*Curious Properties of an Iterative Process   [in MathSource: Packages and Programs]
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IterativeProcess.nb (2.9 MB) - Mathematica Notebook