On A Conjecture Of Borwein And Choi In Short Intervals

Let m≥2 and let Q(X) = X^TAX be a non-singular quadratic form, with discriminantâ–³. Here X = (X₁, X₂,..., X_m)∈Z^m, X^T is the transpose of X, and A = (A_i,j)_{1≤i,j≤m} is a symmetric matrix withâ–³= DetA. Q(X) is said to be indefiniteor positive definite according as whether A is indefinite or positive definite.Given a non-negative integer n, we are concerned with the locus of pointsR(n, Q) := {X = (X₁, X₂,..., X_m)∈Z^m : X≠0, Q(X) = n}.Denote r(n, Q) = #R(n, Q). A basic problem in quadratic form theory is to decide whether or not R(n,Q) is empty, for given n and Q. This problem has been studiedby many authors (see [2], [4], [7], [8], [27], [28]). And moreover, when R(n,Q) is nonempty, our object is to provide upper bounds for r(n,Q). In this direction, [16] established upper bounds for quaternary quadratic forms and [22] proved correspondingresults for ternary quadratic forms.The theory of binary quadratic forms is special; it corresponds to the arithmeticof ideals in an imaginary quadratic number field. General cases began to be considered systematically in the late nineteenth century, and the theory matured in the hands of Minkowski, Hasse, Siegel, to mention a few original contributors. Minkowski created the geometry of numbers to answer the question of equivalence of quadratic forms. His methods of reduction of forms yield practical algorithms for the determination of all solutions. Hasse's work is oriented towards the algebraic aspect. In particular, his local-global principle produces intrinsic conditions for the existence of solutions. Siegel developed analytic methods to find, among many other things, a formula for the number of representations. He is also credited with the extensive use and development of automorphic forms in this connection.In this paper, we consider a special case, the quadratic definite quadratic form Q(X,Y) = X² + NY², where N is a positive squarefree integer. Let r_N(n) denote the number of integer solutions to Q(X,Y) = n (counting signs and order). We gives an estimate of the quantitywhenever x^3/4≤y≤x.