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2004 Approximate Squaring
J. C. Lagarias, N. J. A. Sloane
Experiment. Math. 13(1): 113-128 (2004).


We study the "approximate squaring'' map {\small $f(x) := x \lceil x \rceil$} and its behavior when iterated. We conjecture that if f is repeatedly applied to a rational number {\small $r = l/d > 1$} then eventually an integer will be reached. We prove this when {\small $d=2$}, and provide evidence that it is true in general by giving an upper bound on the density of the "exceptional set'' of numbers which fail to reach an integer. We give similar results for a p-adic analogue of f, when the exceptional set is nonempty, and for iterating the "approximate multiplication'' map {\small $f_r(x) := r \lceil x \rceil$}, where r is a fixed rational number. We briefly discuss what happens when "ceiling'' is replaced by "floor'' in the definitions.


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J. C. Lagarias. N. J. A. Sloane. "Approximate Squaring." Experiment. Math. 13 (1) 113 - 128, 2004.


Published: 2004
First available in Project Euclid: 10 June 2004

zbMATH: 1115.11045
MathSciNet: MR2065571

Primary: 26A18
Secondary: 11B83 , 11K31 , 11Y99

Keywords: 3x + 1 problem , approximate multiplication , Approximate squaring , integer sequences , iterated maps , Mahler z-numbers

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 1 • 2004
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