منابع مشابه
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...
متن کاملSystems of quadratic diophantine inequalities
has a nonzero integer solution for every > 0. If some Qi is rational and is small enough then for x ∈ Zs the inequality |Qi(x)| < is equivalent to the equation Qi(x) = 0. Hence if all forms are rational then for sufficiently small the system (1.1) reduces to a system of equations. In this case W. Schmidt [10] proved the following result. Recall that the real pencil generated by the forms Q1, . ...
متن کاملOn the Frobenius Number of a Modular Diophantine Inequality
We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality ax mod b 6 x, with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.
متن کاملInteger-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 ...
متن کاملLagrange, 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...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1961
ISSN: 0002-9939
DOI: 10.2307/2034145