相关论文: The unit equation has no solutions in number field…
Representative examples of our results are as follows. For any positive integer $N$ the equation $$ x^3+y^3=z^3+t^3, \quad x,y,z,t\in \mathbb{N}, \quad \{x,y\}\not=\{z,t\} $$ has no solutions satisfying $$ N\le x,y,z,t <…
We show that there exists an atomic representable polyadic equality algebra of finite dimension n\geq 3, such that the cylindric reduct of its completion is not in SNr_n\CA_{n+4}, hence the result in the title. This solves an open problem…
Kelly's theorem states that a set of $n$ points affinely spanning $\mathbb{C}^3$ must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least…
A class number formula is proved for extended ring class fields $L_{\mathcal{O},9}$ over imaginary quadratic fields $K_d = \mathbb{Q}(\sqrt{-d})$, in which the prime $p = 3$ splits, by determining the fields generated by the periodic points…
We study the semilinear elliptic system \[ \Delta u = p(|x|)\,g(v), \qquad \Delta v = q(|x|)\,f(u), \qquad x \in \mathbb{R}^n,\; n \geq 3, \] under new Keller--Osserman-type integral conditions on the nonlinearities $f,g$ and decay…
We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in…
Let $S=\{x^2+c_1, x^2+c_2,\dots, x^2+c_s\}$ be a set of quadratic polynomials with rational coefficients, and let $P$ be a rational basepoint. We classify the pairs $(S,P)$ for which $P$ has finite orbit for $S$, assuming that the maximum…
For an abelian totally real number field $F$ and an odd prime number $p$ which splits totally in $F$, we present a functorial approach to special "$p$-units" previously built by D. Solomon using "wild" Euler systems. This allows us to prove…
Richard Guy asked the following question: can we find a triangle with rational sides, medians, and area? Such a triangle is called a \emph{perfect triangle} and no example has been found to date. It is widely believed that such a triangle…
Given two polynomials $P,q$ we consider the following question: "how large can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ be, assuming the sequence is not identically zero?". The answer $K$ to this question is known…
We prove several results on backward orbits of rational functions over number fields. First, we show that if $K$ is a number field, $\phi\in K(x)$ and $\alpha\in K$ then the extension of $K$ generated by the abelian points in the backward…
Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the…
We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only…
We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric $2s$-stable operators. Firstly, in the whole space…
Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K_{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence…
Static spherically symmetric uncoupled scalar space-times have no event horizon and a divergent Kretschmann singularity at the origin of the coordinates. The singularity is always present so that non-static solutions have been sought to see…
We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field…
In this note we show that it is impossible to have data independent non-oscillatory three point finite difference scheme irrespective of its accuracy for a scalar hyperbolic initial value problem.
Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u…
In the present paper we study symmetries of extended $K(m,n,p)$ equation $$ u_t=a(u^p)_{xxxxx}+b(u^n)_{xxx} + c(u^m)_{x} + f(u), $$ where $a,b,c$ are arbitrary real constants and $m,n,p$ are arbitrary integers, and prove that for $a\neq 0$…