Related papers: A systematic approach to Diophantine equations: op…
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
Monograph "B. Grechuk, Polynomial Diophantine equations. A systematic approach" suggests solving Diophantine equations systematically in certain order. Many hundreds of the equations are left to the reader. Here, we provide complete…
This paper reports on the current status of the project in which we order all polynomial Diophantine equations by an appropriate version of "size", and then solve the equations in that order. We list the "smallest" equations that are…
We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.
Motivated by questions in cryptography, we look for diophantine equations that are hard to solve but for which determining the number of solutions is easy.
In this paper we propose a method of solving a Nonlinear Diophantine Equation by converting it into a System of Diophantine Linear Equations.
These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…
This paper initiates a novel research direction in the theory of Diophantine equations: define an appropriate version of the equation's size, order all polynomial Diophantine equations starting from the smallest ones, and then solve the…
We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
In this paper we give solutions of certain diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th…
To give a parametrization of the Diophantine equation $A^{3}+B^{3}=C^{3}+D^{3}$ in terms of integral binary quadratic forms in a constructive way.
We discuss several open problems in Diophantine approximation. Among them there are famous Littlewood's and Zaremba's conjectures as well as some new and not so famous problems.
In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences…
We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.
We present a general algorithm for solving all two-variable polynomial Diophantine equations consisting of three monomials. Before this work, even the existence of an algorithm for solving the one-parameter family of equations…
Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer solutions and check for…
We consider Diophantine equations of the shape $ f(x) = g(y) $, where the polynomials $ f $ and $ g $ are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many…
In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations $\prod_{i=1}^m z_i^{-1} c_i z_i = 1$ (and its variants) over the class of finite metabelian groups…
In this short note we present a method of solving this Diophantine equation, method which is different from Ljunggren's, Mordell's, and R.K.Guy's.