Related papers: A SAT Attack on the Erdos Discrepancy Conjecture

In 1999 Allan Swett checked (in 150 hours) the Erd\H{o}s-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…

Let $n$ be a positive integer and let $S$ be a sequence of $n$ integers in the interval $[0,n-1]$. If there is an $r$ such that any nonempty subsequence with sum $\equiv 0$ $\pmod n$ has length $=r,$ then $S$ has at most two distinct…

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 Erd\H{o}s conjectured in 1990 that the number of such…

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2.…

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)).

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…

The classical Erd\H{o}s-Szekeres theorem dating back almost a hundred years states that any sequence of $(n-1)^2+1$ distinct real numbers contains a monotone subsequence of length $n$. This theorem has been generalised to higher dimensions…

A sequence $S=s_{1}s_{2}..._{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization…

A famous result by Erd\H{o}s and Szekeres (1935) asserts that, for all $k,d \in \mathbb{N}$, there is a smallest integer $n = g^{(d)}(k)$ such that every set of at least $n$ points in $\mathbb{R}^d$ in general position contains a $k$-gon,…

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known…

Let P=(p_1,p_2,...,p_N) be a sequence of points in the plane, where p_i=(x_i,y_i) and x_1<x_2<...<x_N. A famous 1935 Erdos--Szekeres theorem asserts that every such P contains a monotone subsequence S of $\sqrt N$ points. Another, equally…

The \textit{Collatz's conjecture} is an unsolved problem in mathematics. It is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse conjecture or problem. Take any positive integer $ n $. If $ n $ is even then divide it…

From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that…

Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof…

By a profound result of Heinrich, Novak, Wasilkowski, and Wo{\'z}niakowski the inverse of the star-discrepancy $n^*(s,\ve)$ satisfies the upper bound $n^*(s,\ve) \leq c_{\mathrm{abs}} s \ve^{-2}$. This is equivalent to the fact that for any…

In this paper, by constructing extremely hard examples of CSP (with large domains) and SAT (with long clauses), we prove that such examples cannot be solved without exhaustive search, which is stronger than P $\neq$ NP. This constructive…

Let $D$ be a set of positive integers. A $D$-diffsequence of length $k$ is a sequence of positive integers $a_1 < \cdots < a_k$ such that $a_{i+1}-a_i\in D$ for $i=1,\ldots,k-1$. For $D=\{2^i\mid i\in \mathbb{Z}_{\ge 0}\}$, it is known that…

The Erd\H{o}s-Straus conjecture, proposed in 1948 by Paul Erd\H{o}s and Ernst G. Straus, asks whether the Diophantine equation \[ \frac{4}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b, c, d \in…

We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture…

In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n.…