Let Vn(q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of Vn(q) is a partition of Vn(q) if every nonzero vector in Vn(q) is contained in exactly one subspace of P. If there exists a partition of Vn(q) containing ai subspaces of dimension ni for 1 ≤ i ≤ k, then (ak, ak−1, . . . , a1) must satisfy the Diophantine equation ∑k i=1 ai(q ni − 1) = q− 1. In...

The paper introduces a connectionist network approach to find numerical solutions of Diophantine equations as an attempt to address the famous Hilbert‟s tenth problem. The proposed methodology uses a three layer feed forward neural network with back propagation as sequential learning procedure to find numerical solutions of a class of Diophantine equations. It uses a dynamically constructed net...

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions: (i) Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem? (ii) Is it useful to combine this approach with traditional approaches to Diophantine e...

The rules for unification in a simple syntactic theory, using Kirchner’s mutation [15, 16] do not terminate in the case of associativecommutative theories. We show that in the case of a linear equation, these rules terminate, yielding a complete set of solved forms, each variable introduced by the unifiers corresponding to a (trivial) minimal solution of the (trivial) Diophantine equation where...

It is known that finding the shortest solution for a linear Diophantine equation is a NP problem. In this paper we have devised a method, based upon the basis reduction algorithm, to obtain short solutions to a linear Diophantine equation. By this method we can obtain a short solution (not necessarily the shortest one) in a polynomial time. Numerical experiments show superiority to other method...

In this work, we consider the number of integer solutions of Diophantine equation D : y − 2yx − 3 = 0 over Z and also over finite fields Fp for primes p ≥ 5. Later we determine the number of rational points on curves Ep : y = Pp(x) = y p 1 + y p 2 over Fp, where y1 and y2 are the roots of D. Also we give a formula for the sum of x− and y−coordinates of all rational points (x, y) on Ep over Fp. ...

Diophantine criteria occur naturally in the theory of partial differential equations through the notorious problem of small denominators. An extensive treatment of such problems in the theory of PDEs can be found, e.g., in [6]. In this paper, we are interested in a Diophantine problem related to an inhomogeneous wave equation in n spatial and one temporal dimension with periodic boundary condit...

We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation C„ = wpTM< ■ ■ ■ p"< (where {G„}^_o is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation x2 + k = p\' ■ ■ ■ pf', which is a generalization of the Ramanujan-Nagell equation x2 + 7 = ...

Systems based on the splicing operation are computationally complete. Usually demonstrations of this are based on simulations of type-0 grammars. We propose a diierent way to reach this result by solving Diophantine equations using extended H system with permitting context. Completeness then follows from Matiyasevich's theorem stating that the class of Diophantine sets is identical to the class...