Related papers: The normality of digits in almost constant additiv…

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 25 is the sum of at most three natural numbers whose…

In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…

The binary sum-of-digits function $s$ counts the number of ones in the binary expansion of a nonnegative integer. For any nonnegative integer $t$, T.~W.~Cusick defined the asymptotic density $c_t$ of integers $n\geq 0$ such that…

According to a popular belief, the decimal digits of mathematical constants such as {\pi} behave like statistically independent random variables, each taking the values 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 with equal probability of 1/10. If…

We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The…

By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions:…

Let b > 1 be an integer and denote by s_b(m) the sum of the digits of the positive integer m when is written in base b. We prove that s_b(n!) > C_b log n log log log n for each integer n > e, where C_b is a positive constant depending only…

The equidistribution of the double sign patterns of the Liouville function $\lambda$ is proved unconditionally. As application, it is shown that the computable real number $$\sum_{n \geq 1} \frac{1+\lambda(n)}{2^{n}}$$ is a simply normal…

In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…

Let $k$ and $n$ be natural numbers. Let $\omega_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of…

We derive some new finite sums involving the sequence $s_{2}\left(n\right),$ the sum of digits of the expansion of $n$ in base $2.$ These functions allow us to generalize some classical results obtained by Allouche, Shallit and others.

In this short note, we give a proof, conditional on the Generalized Riemann Hypothesis, that there exist numbers x which are normal with respect to the continued fraction expansion but not to any base b expansion. This partially answers a…

A positive integer $n$ is practical if every $m \leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical numbers by applying an arithmetic function $f$ to each of the divisors of $n$ and…

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with $x,y,z$ positive integers. The \emph{Erd\H{o}s-Straus conjecture} asserts that…

Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a…

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions…

In this article, we will use elementary number theory techniques to investigate a sequence of integers defined by a sifting process called the lucky numbers. Ulam introduced lucky numbers as a sieve-based analogue of prime numbers. We…

We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…

We show the existence of ``good'' approximations to a real number $\gamma$ using rationals with denominators formed by digits $0$ and $1$ in base $b$. We derive an elementary estimate and enhance this result by managing exponential sums.

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^0 x, b^1 x, b^2 x , \ldots)$ is essentially the same as…