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.
In this note, we propose a conjecture stating that some series involving primitive sequences are convergent. Then, we show (by a counterexample) that the analogue of a conjecture of Erdős, for those series, is false.
Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
OpenAI；Queens’ College, University of Cambridge（剑桥大学女王学院）；School of Mathematics, Southeast University（东南大学数学学院）；School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
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.
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 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…
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)).
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}…
A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging 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. One source of difficulty in this conjecture is that $\sum n^{-λ}$ over a primitive set is maximized by the primes if and only if $λ$ is at least the critical exponent $τ_1 \approx 1.14$. A set is $k$-primitive if no member divides any product of up to $k$ other distinct members.…
Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures
Electro-Gravitational Space Propulsion Laboratory (EGSPL)（电引力空间推进实验室）；Maulana Azad National Institute of Technology Bhopal（莫拉纳·阿扎德国家技术学院博帕尔分校）；National Taiwan University（国立台湾大学）
A set of natural numbers is primitive if no element of the set divides another. Erdős conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in ¶} 1/(p log p), where ¶denotes the set of primes. In this paper, we make progress towards this conjecture by restricting the setting to smaller sets of primes. Let P denote any subset of ¶, and let N(P) denote the set of natural numbers all of whose prime factors are in P. We say that P is Erdős-best among primitive subsets of N(P) if the inequality \sum_{n\in S} 1/(n log n) \le \sum_{n\in P} 1/(p log p) holds for every …
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.
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 …
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…
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 $k=2s-1$.
We introduce a variation on the prime divisor function $B(n)$ of Alladi and Erdős, a close relative of the sum of proper divisors function $s(n)$. After proving some basic properties regarding these functions, we study the dynamics of its iterates and discover behaviour that is reminiscent of aliquot sequences. We prove that no unbounded sequences occur, analogous to the Catalan-Dickson conjecture, and give evidence towards the analogue of the Erdős-Granville-Pomerance-Spiro conjecture on the pre-image of $s(n)$.
This paper makes a fundamental assertion about the Erdős-Straus conjecture. Suppose that for a prime $p$ there exists $x,y,z \in \mathbb{N}$ with $x \leq y \leq z$ so that $$ \frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}. $$ The main contribution of this paper is that, under this assumption, the Erdős-Straus conjecture can be reduced by one variable. For example, it is necessarily true that $$ z = \frac{xyp}{\gcd(y,p) \gcd \left( xy, x+y \right)}.$$ Considering other reductions of the Erdős-Straus conjecture, this paper suggests a method for proof.
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…
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.
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 $n\equiv13 \textrm{ mod } 24.$
Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erdös conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved the conjecture. We show that this conjecture is true for $82\%$ of the remaining integers $q\equiv 1$ ( mod $4$).
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 $4/n$.
Erdős conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erdős' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary …
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 $N=10^{17}$.
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…
This paper is a preliminary expository paper that outlines the relationship between solutions to the Erdős-Straus conjecture for a given prime $p$ and their corresponding Pythagorean triples. This paper also uses Bézout Coefficients to find a relationship between the solutions to the conjecture and roots of a second degree polynomial.
In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve functi…
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.
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…
In an attempt to resolve a folklore conjecture of Erdős regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{-1, 1 \}$, Livingston conjectured the $\overline{\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \geq 4$, highlighting that a new approach is required to settles Erdős's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients are needed to settle Erdős's co…
In a written correspondence with A. Livingston, Erdős conjectured that for any arithmetical function $f$, periodic with period $q$, taking values in $\{-1,1\}$ when $q \nmid n$ and $f(n)=0$ when $q \mid n$, the series $\sum_{n=1}^{\infty} f(n)/n$ does not vanish. This conjecture is still open in the case $q \equiv 1 \bmod 4$ or when $2 ϕ(q)+ 1 \leq q$. In this paper, we obtain the characteristic function of the limiting distribution of $L(k,f)$ for any positive integer $k$ and Erdős function $f$ with the same parity as $k$. Moreover, we show that the Erdős conjecture is true with "probability"…
Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote the sequence of all gaps between consecutive primes. In 1948 Erdős and Turán showed that d_{n+1}-d_n changes sign infinitely often and together with Pólya asked for a necessary and sufficient condition that a fixed linear combination of r>1 consecutive prime gaps d_{n+i} i=(1,2,...r) should change sign infinitely often as n runs through the sequence of all natural numbers. They conjectured that the condition is that the non zero values among the coefficients of the linear combination cannot all have the same sign but they…
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…
A rational distance set in the plane is a point set which has the property that all pairwise distances between its points are rational. Erd\H os and Ulam conjectured in 1945 that there is no dense rational distance set in the plane. In this paper we associate an algebraic surface in $\mathbb{P}^3$, that we call a distance surface, to any finite rational distance set in the plane. Under a mild condition, we prove that a distance surface is always a surface of general type. From this, we deduce that the Bombieri-Lang conjecture in arithmetic algebraic geometry (restricted to the classes of surfa…
