Related papers: Ternary expansions of powers of 2

Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any…

The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…

This paper explores whether or not a complete ternary full adder, whose input variables can independently be '0', '1', or '2', is indispensable in the arithmetic blocks of adder, subtractor, and multiplier. Our investigations show that none…

Cantor's diagonal method is traditionally used to prove the uncountability of the set of all infinite binary sequences. This paper analyzes the expressive limits of this method. It is shown that under any constructive application --…

We examine the asymptotic expansion of the Touchard polynomials $T_n(z)$ (also known as the exponential polynomials) for large $n$ and complex values of the variable $z$. In our treatment $|z|$ may be finite or allowed to be large like…

The odd part of 2^e! as e approaches infinity leads to a 2-adic integer z. The bits of z were publicized in OEIS-A359349, where two conjectures were made, relevant to computing z. We prove both of those conjectures. A second 2-adic integer,…

We study the relationship between the frequency of a ternary digit in a number and the asymptotic mean value of the digits. The conditions for the existence of the asymptotic mean of digits in a ternary number are established. We indicate…

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of…

A $d$-dimensional zero-one matrix $A$ avoids another $d$-dimensional zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeroes. Let $f(n,P,d)$ denote the maximum number of ones in a $d$-dimensional…

Motivated by recent results of Kapron and Steinberg (LICS 2018) we introduce new forms of iteration on length in the setting of applied lambda-calculi for higher-type poly-time computability. In particular, in a type-two setting, we…

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a…

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that…

This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…

Real and imaginary part of the limit 2N->infinity of the integral int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over…

Let $\{s_n\}$ and $\{t_n\}$ be two sequences of positive real numbers. Under some mild conditions on $\{s_n\}$ and $\{t_n\}$, we give the precise formula of the Hausdorff dimension of the set \[ \mathbb{E}(\{s_n\},\{t_n\}):=\Big\{x\in(0,1):…

A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erd\H{o}s in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the…

We introduce a method for deriving constraints on the symbol of Feynman integrals from the form of their asymptotic expansions in the neighborhood of Landau loci. In particular, we show that the behavior of these integrals near singular…

Let $\Lambda$ be the limit set of an infinite conformal iterated function system and let $F$ denote the set of fixed points of the maps. We prove that the box dimension of $\Lambda$ exists if and only if \[ \overline{\dim}_{\mathrm B} F\leq…

E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like…

The digits of Pierce expansion obey the law of large numbers, the central limit theorem, and the law of the iterated logarithm in the Lebesgue measure sense. We calculate the Hausdorff dimensions of the exceptional sets of each of the three…