Related papers: Catalan's Conjecture over Number Fields

This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples…

Motivated by their research on automorphism groups of pseudo-real Riemann surfaces, Bujalance, Cirre and Conder have conjectured that there are infinitely many primes $p$ such that $p+2$ has all its prime factors $q\equiv -1$ mod~$(4)$. We…

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is…

Erd\H{o}s conjectured in 1956 that there are $x^{1-o(1)}$ Carmichael numbers up to $x$. Pomerance made this conjecture more precise and proposed that there are $x^{1-{\frac{\{1+o(1)\}\log\log\log x}{\log\log x}}}$ Carmichael numbers up to…

We continue the study of the rational-slope generalized $q,t$-Catalan numbers $c_{m,n}(q,t)$. We describe generalizations of the bijective constructions of J. Haglund and N. Loehr and use them to prove a weak symmetry property…

Assuming Calabi symmetry, we prove that a numerical condition ensures the solvability of the complex Hessian quotient equation, as conjectured by Sz\'ekelyhidi. We also propose a conjecture on the existence of a $k$-subharmonic…

This article provides a thorough investigation into Gilbert's Conjecture, pertaining to Hardy spaces in the upper half-space valued in Clifford modules. We explore the conjecture proposed by Gilbert in 1991, which seeks to extend the…

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…

We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -\kappa^2$ for some $\kappa\geq 0.$ We first prove a McKean-type…

In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of…

In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson's conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson's conjecture to the higher order integral polynomial case. However, they did not…

By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this…

A classical problem due to Abel is to determine if a differential equation $y'=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta$ is a given algebraic function over $\mathbb C(x)$. Risch designed an…

In 1995, Nadirashvili and subsequently Ashbaugh and Benguria proved the Rayleigh Conjecture concerning the first eigenvalue of the bilaplacian with clamped boundary conditions in dimension $2$ and $3$. Since then, the conjecture has…

In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was…

In 1898 Charles Jean de la Valle Poussin, as part of his famed proof of the prime number theorem, developed an ineffective bound on the first Chebyshev function of the form: \[ |\theta(x)-x| = \mathcal{O}\left(x \exp(-K \sqrt{\ln…

In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over $\mathbb{Q}$ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence…

A conjecture of Min\'a\v{c} and T\^an predicts that for any n>2, any prime p and any field k, the Massey product of n Galois cohomology classes in H^1(k,Z/pZ) must vanish if it is defined. We establish this conjecture when k is a number…

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. L. Shapiro [9] found the Chung-Feller properties for the Motzkin paths.…

Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$…