相关论文: A geometric proof that $e$ is irrational and a new…

We show the first known example for a pattern $q$ for which $\lim_{n\to \infty} \sqrt[n]{S_n(q)}$ is not an integer. We find the exact value of the limit and show that it is irrational. Then we generalize our results to an infinite sequence…

In this paper, we present a result on using algebraic conjugates to form a sequence of approximations to an algebraic number, and in this way obtain effective irrationality measures for related algebraic numbers. From this result, we are…

Measures of irrationality are a numerical way of quantifying how far a given variety is from being rational (or rationally connected, uniruled, etc.). In the last two decades, there has been renewed interest in the study of these…

Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

A famous theorem of Zudilin states that at least one of the Riemann zeta values $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result,…

We discuss arithmetic questions related to the "poor man's ad\`ele ring" $\mathcal A$ whose elements are encoded by sequences $(t_p)_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $\mathbb Z/p\mathbb Z$. Our main…

We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution.

In this short paper we show that the inequality of arithmetic and geometric means is reduced to another interesting inequality, and a proof is provided.

Let s and k be integers with s \geq 2 and k \geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\ell)…

In this paper, we show how geometry plays in the study of the Furstenberg conjecture (refer to~\cite{F}). Let $p>1$ and $q>1$ be two relative prime positive integers. We prove that a non-atomic $p$- and $q$-invariant measure having balanced…

Let $m>2$ and $q>0$ be integers with $m$ even or $q$ odd. We show the supercongruence $$\sum_{k=0}^{p-1}(-1)^{km}\binom{p/m-q}{k}^m\equiv0\pmod{p^3}.$$ for any prime $p>mq$. This confirms a conjecture of Sun.

Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is…

In this paper, we define the deformed Euler $(s,t)$-numbers ${\rm e}_{s,t,u}$ Furthermore, we prove that ${\rm e}_{as,a^2t,u^{-1}}$ and ${\rm e}_{as,a^2t,u^{-1}}^{-1}$ are irrational numbers when $a,u\in\mathbb{Q}$ and $\vert au\vert>1$,…

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We prove that, if the irrationality exponent of $\xi$ is equal to $2$ or slightly greater than $2$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable…

In this paper we show that certain sets are dense in $\mathbb{R}$. We give some applications. For example, we show an analytical proof that $q^{\frac{1}{n}}$, $q$ is a prime number and $e$; are irrational numbers. As another application we…

For a set $G$ of points in $\PG(m-1,q)$, let $\ex_q(G;n)$, denote the maximum size of a collection of points in $\PG(n-1,q)$ not containing a copy of $G$, up to projective equivalence. We show that \[\lim_{n\rightarrow \infty}…

To account for the first proof of existence of an irrational magnitude, historians of science as well as commentators of Aristotle refer to the texts on the incommensurability of the diagonal in Prior Analytics, since they are the most…

Let $\xi$ be an irrational algebraic real number and $(p_k / q_k)_{k \ge 1}$ denote the sequence of its convergents. Let $(u_n)_{n \geq 1}$ be a non-degenerate linear recurrence sequence of integers, which is not a polynomial sequence. We…

We show how the recent isogeny bounds due to \'E. Gaudron and G. R\'emond allow to obtain the triviality of X_0^+ (p^r)(Q), for r>1 and p a prime exceeding 2.10^{11}. This includes the case of the curves X_split (p). We then prove, with the…