Ben Green
A set A of integers is said to be sum-free if there are no solutions to the equation x + y = z with x,y and z all in A. Answering a question of Cameron and Erdos, we show that the number of sum-free subsets of {1,...,N} is O(2^(N/2)).
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We propose a generalisation of the Cameron-Erdos conjecture for sum-free sets to arbitrary non-translation invariant linear equations over Z in three or more variables and, using well-known methods from graph theory, prove a weak form of the conjecture for a class of equations where the structure of the maximum-size sets avoiding solutions to the equation has been previously obtained.
Let $f(n)$ count the number of subsets of $\{1,...,n\}$ without an element dividing another. In this paper I show that $f(n)$ grows like the $n$-th power of some real number, in the sense that $\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists. This confirms a conjecture of Cameron and Erdös, proposed in a paper where they studied a number of similar problems, including the well known "Cameron-Erdös os Conjecture" on counting sum-free subsets.
In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erdős conjectured in 1990 that the number of such sets is $O(2^{n/2})$. This conjecture was confirmed by Green and, independently, by Sapozhenko. Here we prove a refined version of their theorem, by showing that the number of sum-free subsets of $[n]$ of size $m$ is $2^{O(n/m)} {\lceil n/2 \rceil \choose m}$, for every $1 \le m \le \lceil n/2 \rceil$. For $m \ge \sqrt{n}$, this result is sharp up to the constant…
A set of integers greater than 1 is primitive if no element divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erdős question in the affirmative for 2-primitive sets. Here a set is 2-primitive if no element divides the product of 2 other elements.
Let A_1,...,A_n be finite subsets of a field F, and let f(x_1,...,x_n)=x_1^k+...+x_n^k+g(x_1,...,x_n)\in F[x_1,...,x_n] with deg g<k. We obtain a lower bound for the cardinality of {f(x_1,...,x_n): x_1\in A_1,...,x_n\in A_n, and x_i\not=x_j if i\not=j}. The result extends the Erdos-Heilbronn conjecture in a new way.
A conjecture of Erdős on $p+2^k$
本文研究了不能表示为素数与2的幂之和的正奇数集合，证明其不能表示为有限多个无限等差数列与渐近密度零集合的并集，从而否定了P. Erdős的猜想。
Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures
本文证明了 Erdős-Ulam 关于单色并封闭族的两个猜想：任何二染色中均存在大小超多项式增长的单色并封闭族，且某些染色无指数大小族，结果对任意颜色数成立并已在 Lean 中验证。
The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements $a_1,...,a_n$ of F, the set {a_1x_1+...+a_nx_n: x_1,....,x_n are distinct elements of A} has cardinality at least min{p(F)-delta, n(|A|-n)+1}, where the additive order p(F) of the multiplicative identity of F is different from n+1, and delta=0,1 takes the value 1 if and only if n=2 and $a_1+a_2=0$. In this pap…
Let A be a set of nonnegative integers. For every nonnegative integer n and positive integer h, let r_{A}(n,h) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,..., a_h are elements of A and a_1 \leq a_2 \leq ... \leq a_h. The infinite set A is called a basis of order h if r_{A}(n,h) \geq 1 for every nonnegative integer n. Erdos and Turan conjectured that limsup_{n\to\infty} r_A(n,2) = \infty for every basis A of order 2. This paper introduces a new class of additive bases and a general additive problem, a special case of which is the Erdos-Turan …
Cameron and Erdős raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal sum-free sets in $\{1, \dots , n\}$. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, Sós and Temkin on the structure of sum-free sets.
Since their introduction by Erdős in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. In particular, Erdős asked if the moduli can be distinct and all arbitrarily large, Erdős and Selfridge asked if the moduli can be distinct and all odd, and Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. Another beautiful conjecture, propo…
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.
Paul Erdos conjectured that for every n in N, n>1, there exist a, b, c natural numbers, not necessarily distinct, so that 4/n=1/a+1/b+1/c (see \cite{rg}). In this paper we prove an extension of Mordell's theorem and formulate a conjecture which is stronger than Erdos' conjecture.
A new tool is introduced, specially tailored for the study of order 2 bases of N. It is called the Erdos-Tuan algebra since it suits well their conjecture.
Erdös and Turán once conjectured that any set $A\subset\mathbb{N}$ with $\sum_{a\in A}{1}/{a}=\infty$ should contain infinitely many progressions of arbitrary length $k\geq3$. For the two-dimensional case Graham conjectured that if $B\subset \mathbb{N}\times\mathbb{N}$ satisfies $$\sum\limits_{(x,y)\in B}\frac{1}{x^2+y^2}=\infty,$$ then for any $s\geq2$, $B$ contains an $s\times s$ axes-parallel grid. In this paper it is shown that if the Graham conjecture is true for some $s\geq2$, then the Erdös-Turán conjecture is true for $k=2s-1$.
A short proof of Erdős's $B+C$ conjecture
本文通过简化的证明方法，展示了具有正上Banach密度的自然数集包含两个无限集的和，改进了之前的证明。
This is an exposition of the following `weak' Erdős-Szemerédi conjecture for integer sets proved by Bourgain and Chang in 2004. For any $γ> 0$ there exists $Λ(γ) > 0$ such that for an arbitrary $A \subset \mathbb{N}$, if $|AA| \leq K|A|$ then $$E_{+}(A) \leq K^Λ|A|^{2+γ}.$$
Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erdős conjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently large and $a_1,\dots , a_t$ are positive integers with $a_1<a_2<\cdot\cdot\cdot<a_t\leqslant x$ and $t>\log x$, then there exists an integer $n$ so that the number of solutions of $n=p+a_i$ $(p\in \mathcal{P}, 1\le i\le t)$ is greater than $c$. In this note, we confirm this old conjecture of Erdős.
In 1930s Paul Erdos conjectured that for any positive integer C in any infinite +1 -1 sequence (x_n) there exists a subsequence x_d, x_{2d}, ... , x_{kd} for some positive integers k and d, such that |x_d + x_{2d} + ... + x_{kd}|> C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C=1 a human proof of the conjecture exists; for C=2 a bespoke computer program had generated sequences of length 1124 having discrepancy 2, but the status of the conjecture remained open even for such a small bound…
A strictly increasing sequence $\mathscr{A}$ of positive integers is said to be primitive if no term of $\mathscr{A}$ divides any other. Erdős showed that the series $\sum_{a \in \mathscr{A}} \frac{1}{a \log a}$, where $\mathscr{A}$ is a primitive sequence different from $\{1\}$, are all convergent and their sums are bounded above by an absolute constant. Besides, he conjectured that the upper bound of the preceding sums is reached when $\mathscr{A}$ is the sequence of the prime numbers. The purpose of this paper is to study the Erdős conjecture. In the first part of the paper, we give two sig…
The Erdös--Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x,y,z$ for every postive integers $n\geq 2$. In this short note we find explicity the solutions of the famous conjecture for the case $n\equiv13 \textrm{ mod } 24.$
A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's Conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf, and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.
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 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 $4/n$.
A proof of Erdős's $B+B+t$ conjecture
该研究证明了Erdős的$B+B+t$猜想，证实正上密度自然数集平移后可包含其无限子集对应的限制和集，推进了加性组合领域相关数论问题的研究。
A proof of the Erdős primitive set conjecture
本文证明了 Erdős 本原集猜想，即级数 $f(A) = \sum_{a\in A}1/(a \log a)$ 的上界在素数集合上取得，并将该方法应用于推进多个相关的数论问题。
There are no square $L^2$-flat sequences of polynomials of the type $$\frac{1}{\sqrt q}( ε_0 + ε_1z + ε_2z^2 + \cdots + ε_{q-2}z^{q-2} +ε_q z^{q-1}),$$ where for each $j,~~ 0 \leq j\leq q-1,~ε_j = \pm 1$. It follows that Erdös's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only finitely many Barker sequences. We further get that the spectrum of dynamical systems arising from continuous Morse sequences is singular. This settles an old question due to M. Keane. Applying our reasoning to the Liouville function we obtain that the po…
The Erdös-Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x, y, z$ for every postive integers $n\ge 2$. We generalize the Erdös-Straus equation, state several methods for obtaining well-defined equations, and some features of well-defined equations. Using well-defined equations, we explain the inefficiency of the solutions that did not lead to a complete proof of the problem. Finally, we prove that this conjecture cannot be completely proven by the methods expressed in various articles. For this reason, we express con…
In 1999 Allan Swett checked (in 150 hours) the Erdős-Straus conjecture up to $N=10^{14}$ with a sieve based on a single modular equation. After having proved the existence of a "complete" set of seven modular equations (including three new ones), this paper offers an optimized sieve based on these equations. A program written in C++ (and given elsewhere) allows then to make a checking whose running time, on a typical computer, range from few minutes for $N=10^{14}$ to about 16 hours for $N=10^{17}$.
An Unexpected Connection Between the Discrete Zeta Function and the Erdos-Straus Conjecture Under Mballa's Conjecture
本文在 Mballa 猜想下，将 $4(ζ(s)-1)$ 分解为由 Erdős–Straus 解构成的三项单位分数级数，从而建立离散 zeta 函数与埃及分数问题的联系。
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.
Resolution of two conjectures by Erdős and Hall concerning separable numbers
本文证明了2的幂中可分离数的下密度为正，且不可分离的2的幂的下密度也为正，并证明了乘积等于前k个素数乘积的互锁对数量有限，从而解决了Erdős和Hall的两个猜想。
Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) = \text{max}_{A \in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erdős conjectured that $f(n, k) = |E(n,k)|$ for every $n \ge p_{k}$. Recently Chen and Z…
A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts, and show a connection to certain prime number "races" such as the race between $π(x)$ and li$(x)$.
A Complete Congruence System for the Erdos-Straus Conjecture
本研究围绕Erdos-Straus猜想展开，通过解的结构定义了涵盖所有素数的同余系统，提出A型、B型两类新解并关联对应同余式与多项式，拓展了该猜想的相关研究结果。
On an Erdős-type conjecture on $\mathbb{F}_q[x]$
本文借鉴整数环上Erdős同余覆盖猜想的经典证明方法，将该猜想推广至有限域多项式环$\mathbb{F}_q[x]$，证明了覆盖所有次数小于n的多项式的n个理想陪集必覆盖整个多项式环。
The convergence of an alternating series of Erdős, assuming the Hardy--Littlewood prime tuples conjecture
本文针对Erdős提出的某交错级数收敛性的公开问题，结合素数随机筛选模型与Gallagher计算变体，在强版本Hardy–Littlewood素数元组猜想下证明了该级数条件收敛。
Erdős proved that $\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that $\mathcal{F}(\mathcal{P}_1) > \ldots > \mathcal{F}(\mathcal{P}_k) > \mathcal{F}(\mathcal{P}_{k+1}) > \ldots$, where $\mathcal{P}_j$ is the set of integers with $j$ prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field $\mathbb{F}_q[x]$, investigating the sum $\mathcal{F}…
For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The celebrated Erdős-Turán conjecture says that, if $R_A(n)\ge 1$ for all sufficiently large integers $n$, then the representation function $R_A(n)$ cannot be bounded. For any positive integer $m$, Ruzsa's number $R_m$ is defined to be the least positive integer $r$ such that there exists a set $A\subseteq \mathbb{Z}_m$ with $1\le R_A(n)\le r$ for all $n\in \mathbb{Z}_m$. In 2008, Chen proved that $R_{m}\le 288$ for all positive integers $m$. In thi…
Powers of 3 with few nonzero bits and a conjecture of Erdős
本文使用初等同余方法，找到了所有可表示为至多22个不同2的幂之和的3的幂，以及至多25个不同3的幂之和的2的幂，后者与Erdős猜想有关，并指出初等同余技术对某些指数丢番图方程有效。
A Divisor Parametrization for the Erdős--Straus Conjecture
本文引入基于除数的函数 fab(n,a,b) 来参数化所有分母被 4 整除的 1/n 的三项埃及分数分解，并与已知的 Type I/II 描述进行比较。
On an Erdős--Kac-type conjecture of Elliott
本文证明了Elliott基于Erdős--Kac定理提出的一个关于素数求和渐近性的猜想可由Bombieri--Vinogradov定理推出，并利用素数更深层的分布水平结果证明了一个涉及Poisson--Dirichlet分布的相关结论。
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