## Experimental Mathematics

- Experiment. Math.
- Volume 13, Number 1 (2004), 113-128.

### Approximate Squaring

J. C. Lagarias and N. J. A. Sloane

#### Abstract

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.

#### Article information

**Source**

Experiment. Math., Volume 13, Number 1 (2004), 113-128.

**Dates**

First available in Project Euclid: 10 June 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1086894093

**Mathematical Reviews number (MathSciNet)**

MR2065571

**Zentralblatt MATH identifier**

1115.11045

**Subjects**

Primary: 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25]

Secondary: 11B83: Special sequences and polynomials 11K31: Special sequences 11Y99: None of the above, but in this section

**Keywords**

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

#### Citation

Lagarias, J. C.; Sloane, N. J. A. Approximate Squaring. Experiment. Math. 13 (2004), no. 1, 113--128. https://projecteuclid.org/euclid.em/1086894093