Related papers: Counterexamples to a Conjecture by Alaoglu and Erd…

We give a more strong heuristic justification of our conjecture on the excess of the odious primes.

I discuss the computational methods behind the formulation of some conjectures related to variants on Andrews' $q$-Dyson conjecture.

In this paper, we investigate a number of $q$-supercongruences on double and triple sums. By means of a lemma devised by El Bachraoui and its generalization, we transform some $q$-supercongruences on double and triple sums into the…

We have primarily obtained three results on numbers of the form $p + 2^k$. Firstly, we have constructed many arithmetic progressions, each of which does not contain numbers of the form $p + 2^k$, disproving a conjecture by Erd\H{o}s as Chen…

We consider two recent conjectures of Harrington, Henninger-Voss, Karhadkar, Robinson and Wong concerning relationships between the sum index, difference index and exclusive sum number of graphs. One conjecture posits an exact relationship…

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general…

We confirm several conjectures of Guo, Jouhet and Zeng concerning the factors of alternative binomials sums.

A proof of Sendov's conjecture is given.

In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.

In this note we give a counterexample to a conjecture proposed by Ciliberto about special linear systems of P^n through multiple base points.

We provide a counterexample to P.~Olver's freeness conjecture for $C^\omega$ transformations.

There has been recent interest in a hybrid form of the celebrated conjectures of Hardy-Littlewood and of Chowla. We prove that for any $k,\ell\ge1$ and distinct integers $h_2,\ldots,h_k,a_1,\ldots,a_\ell$, we have $$\sum_{n\leq…

We prove that if a set is `large' in the sense of Erd\H{o}s, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap…

Using a ${}_7F_6$ hypergeometric transformation formula, we prove two supercongruences. In particular, one of these supercongruences confirms a recent conjecture of Guo, Liu and Schlosser, and gives an extension of a supercongruence of Long…

Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erd\H{o}s 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…

In this paper, by making use of one of Chen's theorems and the method of mathematical analysis, we refine Edwards-Child's inequality and solve a conjecture posed by Liu.

The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…

In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these…

We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.

We propose several Hodge theoretic analogues of the conjectures of Hopf and Singer, and prove them in some special cases.