In this paper, we shall prove, for any m ≥ 1, the existence of an uncountable subset of U-numbers of type ≤ m (which we called the set of m-ultra numbers) for which there exists uncountably many transcendental analytic functions mapping it into Liouville numbers.
In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].
In this note, we prove that for any ${\it\nu}>0$, there is no lacunary entire function $f(z)\in \mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$ and $\text{den}f(p/q)\ll q^{{\it\nu}}$, for all sufficiently large $q$.
Revista Colombiana de MatemáticasVolumen 50(2016)2, páginas [139][140][141][142][143] A new proof of the Unique Factorization of Z 1+ 3, 7, 11, 19, 43, 67, 163 Una nueva demostración de la factorizaciónúnica Z 1+ √ −d 2 para Abstract. In this paper, we give an elementary proof of the fact that the rings Z 1+are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.Resumen. En este artículo, damos una demostración elemental de que los anillos Z 1+
√ −d 2Palabras y frases clave. Dominio de factorizaciónúnica, primo, irreducible.
In 1906, Maillet proved that given a non-constant rational function f, with rational coefficients, if is a Liouville number, then so is fðÞ. Motivated by this fact, in 1984, Mahler raised the question about the existence of transcendental entire functions with this property. In this work, we define an uncountable subset of Liouville numbers for which there exists a transcendental entire function taking this set into the set of the Liouville numbers.
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