Related papers: On representations by Egyptian fractions

We improve the known upper bound for short exponential sums and increase the range on which a sharp upper bound is known.

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a series of positive rational terms is a transcendental number.…

A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^3$. As a…

We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.

We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.

In this short note, we compute higher extension groups for all irreducible representations and deduce the multiplicity formula for finite length representations in triple product case.

We provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the…

We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and…

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a sum, product and quotient of some series of positive rational…

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper…

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…

Lower bounds for some explicit decision problems over the complex numbers are given.

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erd\H{o}s-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than…

We prove tight upper bounds for the number of vertices of a simple polygon that is the union or the intersection of two simple polygons with given numbers of convex and concave vertices. The similar question on graphs of the lower (or…

In a base phi representation a natural number is written as a sum of powers of the golden mean $\varphi$. There are many ways to do this. How many? Even if the number of powers of $\varphi$ is finite, then any number has infinitely many…

We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of…

We examine the effect of numerical integration on the convergence of high order pyramidal finite element methods. Rational functions are indispensable to the construction of pyramidal interpolants so the conventional treatment of numerical…

We give an optimal bound for the remainder when counting the number of rational points on the $n$-dimensional sphere with bounded denominator for any $n\geq 2$.

Given a holonomic sequence $F(n)$, we characterize rational functions $r(n)$ so that $r(n)F(n)$ can be summable. We provide upper and lower bounds on the degree of the numerator of $r(k)$ and show the denominator of $r(n)$ can be read from…