Related papers: Dejean's conjecture holds for n>=27

A problem in zero-sum theory is to determine all pairs $(k,n)$ for which every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While all other cases have been solved more than a decade ago, the case when $k$ equals $4$ and…

In number theory, the Erdos-Straus conjecture states that for all n >=2, the rational number 4/n can be expressed as the sum of three unit fractions. Paul Erdos and Ernst G. Straus formulated the conjecture in 1948. The restriction that the…

The Polignac's Conjecture, first formulated by Alphonse de Polignac in 1849, asserts that, for any even number M, there exist infinitely many couples of prime numbers P, P+M. When M = 2, this reduces to the Twin Primes Conjecture. Despite…

We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong…

We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture…

We show that the "majority is least stable" conjecture is true for $n=1$ and $3$ and false for all odd $n\geq 5$.

Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…

We prove Manin's conjecture for a singular cubic surface S with a singularity of type E6. If U is the open subset of S obtained by deleting the unique line from S, then the number of rational points in U with anticanonical height bounded by…

We make effective $l^2 L^p$ decoupling for the parabola in the range $4 < p < 6$. In an appendix joint with Jean Bourgain, we apply the main theorem to prove the conjectural bound for the sixth-order correlation of the integer solutions of…

In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new…

Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevich's conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild…

The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd,…

We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no $n$-by-$m$ systems…

Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of…

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…

For p=5,7, we show that a subset \(E \subset \F_p^3\) is spectral if and only if E tiles \(\F_p^3\) by translation. Additionally, we give an alternate proof that the conjecture holds for p=3.

We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly…

It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is a multiple of 4.

We establish the exact overlaps conjecture for iterated functions systems on the real line with algebraic contractions and arbitrary translations.

This note is to study Bourgain's slicing problem following the routes investigated in the last decade. We show that the slicing constant $L_n$ is bounded by $C\log(\log n) $, $n\geq 3$, for some universal constant $C$.