Related papers: The Dixmier conjecture and the shape of possible c…

In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where…

In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward…

We report the finding of the new upper bound on the lowest positive integer $x$ for which the Mertens conjecture \begin{equation*} \left| \sum_{1 \leq n \leq x} \mu(n) \right| < \sqrt{x} \end{equation*} fails to hold: $x < \exp(1.017 \times…

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a…

Zhi-Wei Sun conjectures if k congruence classes are disjoint, then two of the moduli have greatest common divisor at least as large as k. We prove this conjecture for k strictly less than 21.

We prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the largest modulus is at least 108. We also prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the least…

The purpose of this article is to formulate a number of probabilistic hidden-variable theorems, to provide proofs in some cases, and counterexamples to some conjectured relationships. The first theorem is the fundamental one. It asserts the…

Menger's theorem - the maximum number of vertex-disjoint $X$-$Y$ paths is equal to the minimum size of an $X$-$Y$ separator - is generally not true in bidirected graphs. We prove that Menger's theorem holds true if we take the nontrivial…

Minimal model conjecture for a proper variety $X$ is that if $\kappa(X)\geq 0$, then $X$ has a minimal model with the abundance and if $\kappa =-\infty$, then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y \to Z$…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve…

In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely…

We extend a lower bound of Munshi on sums over divisors of a number $n$ which are less than a fixed power of $n$ from the squarefree case to the general case. In the process we prove a lower bound on the entropy of a geometric distribution…

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding…

From Bombieri's mean value theorem one can deduce the prime number theorem being equivalent to the Riemann hypothesis and the least prime P(q) satisfying P(q)= O(q^2 [ln q]^32) in any arithmetic progressions with common difference q.

We prove, in particular, the well--known Zaremba conjecture from the theory of continued fractions for any prime denominator. More precisely, we show, firstly, that under some mild conditions, for any sufficiently large $q$, there exists…

We tackle the "relative" Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its…

The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the…

The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_1$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P, Q \in A_1$ then $A_1 = K \langle P, Q \rangle$. The Weyl algebra…

This paper proves the minimum size of a supersequence over a set of eight elements is 52. This disproves a conjecture that the lower bound of the supersequence is the partial sum of the geometric Connell sequence. By studying the internal…

We study Brauer's long-standing $k(B)$-conjecture on the number of characters in $p$-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for $p\ge5$ nor in the case of abelian defect.…