相关论文: Higher Descent on Pell Conics II. Two Centuries of…
This article deals with the history of certain aspects of the Pell equation X$^2$ - D y$^2$ = 4. We briefly discuss explicit units, and then study the history of Legendre's equations ax$^2$ - b y$^2$ = 4 with ab=D.
We give a construction of unramified cyclic octic extensions of certain complex quadratic number fields. The binary quadratic form used in this construction also shows up in the theory of 2-descents on Pell conics and elliptic curves, as…
This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.
The aim of this paper is to introduce Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving…
Using the representation of numbers, the methods of decreasing the number of calculation steps for Pell's equation are developed. The formulae relating the natural solution x0 of Pell's equation for A0 with the infinite number of minimal…
We present some questions and suggestion on the second part of the Hilbert 16th problem
Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops…
Some symmetry problems are formulated and solved. New simple proofs are given for the earlier studied symmetry problems.
In this paper, we explore the relationship between repdigits and associated Pell numbers, specifically focusing on two main aspects: expressing repdigits as the difference of two associated Pell numbers, and identifying which associated…
We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary…
We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this…
While looking for abductive explanations of a given set of manifestations, an ordering between possible solutions is often assumed. The complexity of finding/verifying optimal solutions is already known. In this paper we consider the…
The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections…
This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed.…
We introduce a generalization of the continued fraction and Chakravala algorithms for solving the Pell equation, utilizing the LLL-algorithm for rank 2 lattices.
In this paper, we study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We…
Recently, the benefit of heavily overparameterized models has been observed in machine learning tasks: models with enough capacity to easily cross the \emph{interpolation threshold} improve in generalization error compared to the classical…
The Second Parameter (2ndP) Problem recognizes the remarkable role played by horizontal branch (HB) morphology in the development of our understanding of globular clusters, and the Galaxy, over the last 50 years. I will describe the…
This short paper is concerned with polynomial Pell equations \[P^2-DQ^2=1,\] with $P,Q,D\in\Bbb C[X]$ and ${deg}(D)=2$. The main result shows that the polynomials $P$ and $Q$ are closely related to Chebyshev polynomials. We then investigate…
Work in progress concerning alternative formalizations of arithmetic.