                    Title    Curious Properties of an Iterative Process   Authors    Organization: Kwansei Gakuin High School
 Department: Mathematics Department Daisuke Minematsu
 Organization: Osaka University Satoshi Hashiba Munetoshi Sakaguchi Kwansei Gakuin   Revision date    2005-06-10   Description    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.   Subject     Mathematics > Number Theory   Related items     Curious properties of an iterative process under various moduli   Downloads     Iterative Process.nb (277.9 KB) - Mathematica Notebook       