We give a necessary condition for the existence of solutions of the Diophantine equation p = x + ry, with p, q, r distinct odd prime natural numbers. MSC (2000): 11D41,11S15.

In this paper, we prove a theorem about the integer solutions to the Diophantine equation x − q = py, extending previous work of K.Győry, and F.Luca and A.Togbe, and of the author. MSC (2000): 11D41

Diophantine Equations are equations to which only integer solutions or alternatively rational solutions are considered. The study of these equations has fascinated man for thousands of years. The ancient Babylonians enumerated Pythagorean triples, integer solutions to the equation: X2 + y2 = Z 2 The study of Diophantine equations continued through Greece and the Renaissance with Diophantos, Fer...

The rational solutions with as denominators powers of to the elliptic diophantine equation y x x are determined An idea of Yuri Bilu is applied which avoids Thue and Thue Mahler equations and deduces four term S unit equations with special properties that are solved by linear forms in real and p adic logarithms Introduction In a recent paper SW my colleague R J Stroeker and I determined the com...

We investigate positive solutions (x, y) of the Diophantine equation x2 − (k2 + 1)y2 = k2 that satisfy y < k − 1, where k ≥ 2. It has been conjectured that there is at most one such solution for a given k.

This paper deals with one-dimensional (1D) nonlinear Schrödinger equation with a multiplicative potential, subject to Dirichlet boundary conditions. It is proved that for each prescribed integer b > 1, the equation admits smallamplitude quasi-periodic solutions, whose b-dimensional frequencies are small dilation of a given Diophantine vector. The proof is based on a modified infinitedimensional...

For a principal SU(n)-bundle over a compact manifold of dimension 2, 3, 4, we determine the orbit types of the action of the gauge group on the space of connections modulo pointed local gauge transformations. We find that they are given by Howe subgroups of SU(n) for which a certain characteristic equation is solvable. Depending on the base manifold, this equation leads to a linear, bilinear, o...

l=0 |c l | < 1, and xl(z) = x(xl−1(z)) that denote the nth iterate of a map x. In general, F, G are given complex-valued functions of a complex variable. In this paper, analytic solutions of nonlinear iterative functional differential equations are investigated. Existence of locally analytic solutions and their construction is given in the case that all given functions exist regular points. As ...

In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophan...

In this paper we study the Diophantine equations ck(f +42fg+49g) + 28dk(f g + 7fg) = m, where (ck, dk) are solutions of the Pell equation c 2−7d2= 1.