2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134363We study the ``approximate squaring'' map f(x) := x ceiling(x) and its behavior when iterated. We conjecture that if f is repeatedly applied to a rational number r = l/d > 1 then eventually an integer will be reached. We prove this when 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 f_r(x) := r ceiling(x) where r is a fixed rational number.22 pages. Revised Nov 9, 2003: new theorems, including probabilistic interpretation of results, also analogs for floor function (24 pages)Number TheoryInformation TheoryPrimary 26A18; Secondary 11B83, 11K31, 11Y99text