Search

In this paper, we use Baker's theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a "symmetrical" type of numbers) that can be written in the form F n + T n , for some n ≥ 1 , where (F n) n ≥ 0 and (T n) n ≥ 0 are the sequences of Fibonacci and Tribonacci numbers, respectively. [ABSTRACT FROM AUTHOR]

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of e α , for algebraic values of α) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as e γ log   n . This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called γ-th arithmetic zeta function ζ γ (n) : = n γ ( = e γ log   n ), for a positive integer n and a complex number γ. Moreover, we raise a conjecture about the exceptional set of ζ γ , in the case in which γ is transcendental, and we connect it to the famous Schanuel's conjecture. [ABSTRACT FROM AUTHOR]

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C [ x ] , and the problem of finding zeros in Q [ x ] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P (T) Q (T) . In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form P 1 (T) Q 1 (T) ⋯ P n (T) Q n (T) for some transcendental number T, where P 1 , ... , P n , Q 1 , ... , Q n are prescribed, non-constant polynomials in Q [ x ] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of z w when z and w are transcendental. [ABSTRACT FROM AUTHOR]