Values of isotropic quadratic forms at S-integral points

Values of Quadratic Forms at Integral Points : an Elementary Approach

Rational Quadratic Forms and the Local-global Principle

As for most Diophantine equations, quadratic forms were first studied over the integers, meaning that the coefficients aij are integers and only integer values of x1, . . . , xn are allowed to be plugged in. At the end of the 19th century it was realized that by allowing the variables x1, . . . , xn to take rational values, one gets a much more satisfactory theory. (In fact one can study quadra...

Isotropic Vectors of Non-negative Integral Quadratic Forms

On Orbits of SL(2,Z)+ and Values of Binary Quadratic Forms on Positive Integral Pairs

We consider actions of SL(2, ZZ) and SL(2, ZZ)+ (semigroup of matrices with nonnegative integral entries) on the projective space IP and on IP × IP . Results are obtained on orbit-closures under these actions and they are applied to describe a class of binary quadratic forms Q such that the sets Q(ZZ2) or Q(ZZ2 +) are dense in IR. We prove also a result generalising a theorem of Troessaert and ...

2-universal Positive Definite Integral Quinary Diagonal Quadratic Forms

As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms. §