The Quadratic Diophantine Equations x^2− P(t)y^2− 2P′(t)x + 4P(t)y + (P′(t))^2− 4P(t) − 1 = 0

On the Solutions of Quadratic Diophantine Equations

We determine a finite set of representatives of the set of local solutions in a maximal lattice modulo the stabilizer of the lattice in question for a quadratic Diophantine equation. Our study is based on the works of Shimura on quadratic forms, especially [AQC] and [IQD]. Indeed, as an application of the result, we present a criterion (in both global and local cases) of the maximality of the l...

ON THE DIOPHANTINE EQUATION x2+p2k+1 = 4yn

Lagrange, central norms, and quadratic Diophantine equations

As is often the case, some results get rediscovered over time. In particular, some rather striking results of Lagrange are often recreated. For instance, in [6], a result pertaining to the Pell equation for a prime discriminant was recast in the light of nonabelian cohomology groups. Yet, in [1], the authors acknowledged the fact that the result “has been discovered before,” and provided an ele...

THE DIOPHANTINE EQUATION x2+2k =yn, II

New results regarding the full solution of the diophantine equationx2+2k=yn in positive integers are obtained. These support a previous conjecture, without providing a complete proof.

Integer-Valued Quadratic Forms and Quadratic Diophantine Equations

We investigate several topics on a quadratic form Φ over an algebraic number field including the following three: (A) an equation ξΦ · ξ = Ψ for another form Ψ of a smaller size; (B) classification of Φ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the “class” of such a ξ determines a “class” in the orthogonal group of a form Θ such that Φ ≈ Ψ ⊕Θ. Such was done in ...