Related papers: Catalan's Conjecture over Number Fields
In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series,…
During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called `infinite number expressions' and…
In 1930s Paul Erdos conjectured that for any positive integer $C$ in any infinite $\pm 1$ sequence $(x_n)$ there exists a subsequence $x_d, x_{2d}, x_{3d},\dots, x_{kd}$, for some positive integers $k$ and $d$, such that $\mid \sum_{i=1}^k…
Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner \cite{KW}, for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santal\'{o} type inequality (we call it $j$-Santal\'{o} conjecture) for many sets (or more generally for many…
Bolzano and Cantor were the first mathematicians to make significant attempts to measure the size (numerosity) of different infinite collections. They differed in their methodological approaches, with Cantor's prevailing. This led to the…
For a prime $p\equiv 1 \,(\bmod{4})$, let \[ \varepsilon = \frac{1}{2}\left( t + u\sqrt{p}\right) \] be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In 1951, N. Ankeny, E. Artin, and S. Chowla asked whether $p$…
We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Keller's conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We…
The well-known Van Lint--MacWilliams' conjecture states that if $q$ is an odd prime power, and $A\subseteq \mathbb{F}_{q^2}$ such that $0,1 \in A$, $|A|=q$, and $a-b$ is a square for each $a,b \in A$, then $A$ must be the subfield…
We consider non-linear elliptic equations having a measure in the right hand side, of the type $ \divo a(x,Du)=\mu, $ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given,…
An element a in A_n, the Cayley-Dickson algebra is alternative if (a,a,x)=0 for all x. In this paper we characterise such elements for n>3.To do so,we prove first the so called Yui's conjecture:For a and b pure elements in A_n. If (a,x,b)=0…
We initiate a general quantitative study of sets of $\mathcal{M}$-points, which are special subsets of rational points, generalizing Campana points, Darmon points, and squarefree solutions of Diophantine equations. We propose an asymptotic…
The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…
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…
In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the…
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…
The Golomb-Welch conjecture deals with the existence of perfect $e$% -error correcting Lee codes of word length $n,$ $PL(n,e)$ codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper…
Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if…
Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand's postulate could be proven for Carmichael numbers. In this paper, we answer this…
We show that Kaplansky's unit conjecture is true, under the assumption that the underlying field is complex.
The Nullstellensatz, proved by Hilbert in 1893, is a classical result that holds when the base field is algebraically closed. When the base field is finite, a version of Hilbert's Nullstellensatz is given by Terjanian in 1966. Laksov in…