The wish to determine the complete set of rational integral solutions of (1) was expressed by Diaconis and Graham in [2, p. 328]. Apparently, the solutions to this diophantine problem correspond to values of keN for which the Radon transform based on the set of all xeZ\ with exactly four ones is not invertible. We thank Hendrik Lenstra who communicated the problem to Jaap Top, to whom we are eq...

‎in this paper‎, ‎we obtain the dimensions of symmetry classes of polynomials associated with‎ ‎the irreducible characters of the dihedral group as a subgroup of‎ ‎the full symmetric group‎. ‎then we discuss the existence of o-basis‎ ‎of these classes‎.

We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 − N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of ...

In this paper, an explicit formulation for multivariate truncated power functions of degree one is given firstly. Based on multivariate truncated power functions of degree one, an formulation is presented which counts the number of non-negative integer solutions of s × (s + 1)linear Diophantine equations and it can be considered as a multidimensional versions of the formula counting the number ...

We solve the equation xa + xb + 1= yq in positive integers x, y, a, b and q with a > b and q > 2 coprime to φ(x). This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and p-adic logarithms, the hypergeometric method of Thue and Siegel applied p-adically, local methods, and the algorithmic resolution o...

In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we suggest the use of geometric properties of the solution space in finding bounds for searching solu...

We prove that at large disorder, with large probability and for a set of Diophantine frequencies of large measure, Anderson localization in Z is stable under localized time-quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fröhlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanis...

In his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis (“Omega Calculus”) as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations. The object of this paper is to show that partition analysis is ideally suited for being implemented in computer algebra. To this end we have developed the computer algebra ...

We demonstrate that for any well-defined cryptographic protocol, the symbolic trace reachability problem in the presence of an Abelian group operator (e.g., multiplication) can be reduced to solvability of a particular system of quadratic Diophantine equations. This result enables formal analysis of protocols that employ primitives such as Diffie-Hellman exponentiation, products, and xor, with ...