Quadratic Polynomial Happy Functions
Document Type
Article
Publication Title
Rocky Mountain Journal of Mathematics
Publication Date
2023
Abstract/ Summary
Fix a base b and a monic quadratic polynomial function f with nonnegative coefficients. The corresponding quadratic polynomial happy function maps any positive integer to the sum of the images under f of its nonzero digits. We study the behavior of these functions under iteration. Our main result is that for b ≥ 4, given any fixed point or element of a cycle under iterations of this happy function, there exist arbitrarily long arithmetic sequences of positive integers each of which eventually maps to that number. This extends past results for generalized happy functions in a new direction.
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