On definite quadratic forms, which are not the sum of two definite or semi-definite forms

Abstract

be a positive definite quadratic form with determinant D R and integer coefficients a ;Call it an even form if all a ; ; are even ., an odd form if at least one a ;; is odd . Then I,, is called non-decomposable, if it cannot be expressed as a sum of two non-negative quadratic forms with integer coefficients . Mordell 1 ) proved that f, can always be decomposed into a sum of five squares of linear forms with integer coefficients . Ko 2 ) proved that fn can be expressed as a sum of n + 3 integral linear squares, when n=3, 4, 5 . When n = 6, Mordell 3 ) proved that the form