The arithmetic of Fermat curves.
The asymptotic behaviour of the number of solutions of polynomial congruences.
The Brauer-Manin obstruction on a general diagonal quartic surface
The Büchi sequences and Hilbert's Tenth Problem
In this short survey paper we state the Büchi conjecture and discuss its relations with the Hilbert Tenth Problem. We give some generalizations of the conjecture, and include some numerical examples.
The circle method and pairs of quadratic forms
We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.
The classification of pairs of binary quadratic forms
The cubic congruence and binary quadratic forms . II.
The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves
The density of integral points on hypersurfaces of degree at least four
The density of rational points on a pfaff curve
This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.
The density of rational points on cubic surfaces
The density of S-integral points in projective space with respect to a quadric
The difficulty of the local solubility problem for additive equations
The dimensions of integral points and holomorphic curves on the complements of hyperplanes
The Diophantine Equation ...
The Diophantine equation .
The diophantine equation
We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of in relatively prime integers , where , gcd is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation . As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s for...
The Diophantine equation solved via continued fractions.
The Diophantine equation
Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation has only the solution (x,y,z) = (1,1,1). We give an extension of this result.
The Diophantine equation
It is shown that for a given squarefree positive integer D, the equation of the title has no solutions in integers x > 0, m > 0, n ≥ 3 and y odd, nor unless D ≡ 14 (mod 16) in integers x > 0, m = 0, n ≥ 3, y > 0, provided in each case that n does not divide the class number of the imaginary quadratic field containing √(-2D), except for a small number of (stated) exceptions.