We study solvability of the Diophantine equation \begin{equation*} \frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in integers $n, k, a_{1},\ldots, a_{k}$ satisfying the conditions $k\geq 2$ and $a_{i}<a_{i+1}$ for $i=1,\ldots,k-1$. The above Diophantine equation (of polynomial-exponential type) was mentioned in the monograph of Erdős and Graham, where several questions were stated. Some of these questions were already answered by Borwein and Loring. We extend their work and investigate other aspects of Erdős and Graham equation. First of all, we obtain the upper bound for the value $a_{k}$ given in terms of $k$ only. This mean, that with fixed $k$ our equation has only finitely many solutions in $n, a_{1},\ldots, a_{k}$. Moreover, we construct an infinite set $\cal{K}$, such that for each $k\in\cal{K}$, the considered equation has at least five solutions. As an application of our findings we enumerate all solutions of the equation for $k\leq 8$. Moreover, by applying greedy algorithm, we extend Borwein and Loring calculations and check that for each $n\leq 10^4$ there is a value of $k$ such that the considered equation has a solution in integers $n+1=a_{1}<a_{2}<\ldots <a_{k}$. Based on our numerical calculations we formulate some further questions and conjectures.
Erdős, Graham, and Selfridge considered, for each positive integer $n$, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with $n$ is a square. An open problem posed by Granville concerns the size of $t_n$, under the assumption of the ABC Conjecture. We establish some results on the distribution of $t_n$, and in the process solve Granville's problem unconditionally.
Irrationality of rapidly converging series: a problem of Erdős and Graham
Queens’ College, University of Cambridge（剑桥大学皇后学院）；Department of Mathematical Science, Seoul National University（首尔大学数学科学系）；School of Mathematics, Korea Institute for Advanced Study (KIAS)（韩国高等科学研究院数学学院）；University of Zagreb Faculty of Science, Department of Mathematics（萨格勒布大学理学院数学系）；Department of Mathematics, Stanford University（斯坦福大学数学系）
We provide a way to modify and to extend a previously established inequality by P. Erdős, R. Graham and others and to answer a conjecture posed in the nineties by R. Graham, which bears on the lack of divisibility of the central binomial coefficient by three distinct, fixed odd primes. In fact the result will show by using an approach similar to their own which they proved for the case of two fixed odd primes, that the central binomial coefficient is not divisible infinitely often by three distinct and fixed odd primes. Therefore a generalization to more fixed odd primes than three but finite …
In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_q ψ(q)=\infty$ and any number $γ,$ the following set \[ W(ψ,γ)=\{x\in [0,1]: |qx-p-γ|< ψ(q) \text{ for infinitely many } q,p\in\mathbb{N}\} \] has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if $γ$ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced …
In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when $g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an app…
We propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential diophantine equations. We prove that in a sense the principle is valid for "almost all" equations. Based upon this we propose a general method for the solution of exponential diophantine equations. Using a generalization of a result of Erdős, Pomerance and Schmutz concerning Carmichael's $λ$ function, we can make our search systematic for certain moduli needed in the method.
The aim of this note is to show that given a positive integer $n \geq 5$, the positive integral solutions of the diophantine equation $4/n = 1/x + 1/y+1/z$ cannot have solution such that $x$ and $y$ are coprime with $xy < \sqrt{z/2}$. The proof uses the continued fraction expansion of $4/n$.
The Erdös-Moser conjecture states that the Diophantine equation $S_k(m) = m^k$, where $S_k(m)=1^k+2^k+...+(m-1)^k$, has no solution for positive integers $k$ and $m$ with $k \geq 2$. We show that stronger conjectures about consecutive values of the function $S_k$, that seem to be more naturally, imply the Erdös-Moser conjecture.
In this paper, we study the solutions of the equation $F_n-F_m=p^a$ where $p$ is either $7$ or $13$ and $n>m\geqslant 0$, $a\geqslant 2$. We confirm the conjecture of Erduvan and Keskin by proving that there is no solutions for this Diophantine equation. We will use the lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in Diophantine approximation.
In this paper we make a Gaussian integer version of the Erdős-Straus conjecture and we solve the Erdős-Straus diophantine equation over the rings of integers of norm-Euclidean quadratic fields.
In this paper we will explore the solutions to the diophantine equation in the Erdős-Straus conjecture. For a prime $p$ we are discussing the relationship between the values $x,y,z \in \mathbb{N}$ so that $$ \frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.$$ We will separate the types of solutions into two cases. In particular we will argue that the most common relationship found is $$ x = \lfloor \frac{py}{4y-p} \rfloor + 1.$$ Finally, we will make a few conjectures to motivate further research in this area.
The Erdös-Moser equation is a Diophantine equation proposed more than 60 years ago which remains unresolved to this day. In this paper, we consider the problem in terms of divisibility of power sums and in terms of certain Egyptian fraction equations. As a consequence, we show that solutions must satisfy strong divisibility properties and a restrictive Egyptian fraction equation. Our studies lead us to results on the Bernoulli numbers and allow us to motivate Moser's original approach to the problem.
If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.
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{Erdős-Straus conjecture} asserts that $f(n) > 0$ for every $n \geq 2$. To solve this conjecture, it suffices without loss of generality to consider the case when $n$ is a prime $p$. In this paper we consider the question of bounding the sum $\sum_{p<N} f(p)$ asymptotically as $N \to \infty$, where $p$ ranges over primes. Our main result establishes the asymptotic upper and lower bounds $$ N \log^2 N…
An open conjecture of Erdos and Moser is that the only solution of the Diophantine equation in the title is the trivial solution 1+2=3. Reducing the equation modulo k and k^2, we give necessary and sufficient conditions on solutions to the resulting congruence and supercongruence. A corollary is a new proof of Moser's result that the conjecture is true for odd exponents n. We also connect solutions k of the congruence to primary pseudoperfect numbers and to a result of Zagier. The proofs use divisibility properties of power sums as well as Lerch's relation between Fermat and Wilson quotients.
A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A $\le$ B $\le$ C -- 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6. Several estimates on the relative size of the parameters are known, such as the one given by Erd{ö}s C -- B $\le$ 5 log log C, or the one given by Bhat and Ramachandra C --B $\le$ (1/ log 2+o(1)) log log C. We check the conjecture for B $\le$ 10 3000 and give better explicit bounds such as C -- B $\le$ log log(B+1) log 2 -- 0.8803.
Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.
A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to Sylvester's sequence, Giuga numbers, Znám's problem, the inheritance problem, and Curtiss's bound on solutions of a unit fraction equation. Here we show $K \equiv 6 \pmod{6^2}$ if $6\mid K$, and uncover a remarkable $7$-term arithmetic progression of residues modulo $6^2\cdot8$ in the sequence of known PPNs. On that basis, we pose a conjecture which leads to a cond…
Let $n$ be a positive integer. We study the diophantine equation $ab(ab-1)-na=Δ^2$, where $a,b$ are positive integers. We also show that if a system of two congruences is soluble, then an equation which is a translation of Erdős-Straus conjecture is soluble.
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of \$350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdős on the minimum number of colors needed to color the positive integers less than $n$ so that $n$ cannot be written as a monochr…
Let $n$ be a non-negative integer and put $p_{n}(x)=\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\sum_{i=1}^{k}p_{a_{i}}(x), $$ where $m\in\N_{\geq 2}$ and $a_{1}<a_{2}<\ldots <a_{k}<n$. This equation can be considered as a generalization of the Erdős-Selfridge Diophantine equation $y^m=p_{n}(x)$. We present some general finiteness results concerning the integer solutions of the above equation. In particular, if $n\geq 2$ with $a_{1}\geq 2$, then our equation has only finitely many …
Two sets of nonnegative integers $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ are defined as \emph{disjoint}, if $\{A-A\}\bigcap\{B-B\}=\{0\}$, namely, the equation $a_i+b_t=a_j+b_k$ has only trivial solution. In 1984, Erd\H os and Freud [J. Number Theory 18 (1984), 99-109.] constructed disjoint sets $A,B$ with $A(x)>\varepsilon\sqrt{x}$ and $B(x)>\varepsilon\sqrt{x}$ for some $\varepsilon>0$, which answered a problem posed by Erd\H os and Graham. In this paper, following Erdős and Freud's work, we explore further properties for disjoint sets. As a main result, we prove that, for disjoint…
In this article, we continue with our investigation of the Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions $R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\log 3\log\log n$ and variance $(log 3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects.
UVA Department of Mathematics（弗吉尼亚大学数学系）；LIRMM, Univ Montpellier, CNRS（蒙彼利埃大学LIRMM实验室）；MIT Department of Mathematics（麻省理工学院数学系）；UCLA Department of Mathematics（加州大学洛杉矶分校数学系）；Google（谷歌）
Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erdös and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for any sequence of positive integers $n_1,n_2,\ldots$ with $\frac{n_{k+1}}{n_k}\geq c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when $\frac{n_{k+1}}{n_k}\geq c>2$. The bound $c>2$ is best possible thanks to…
