by a positive-definite integral binary quadratic form By

In this note we prove an elementary result concerning the distribution of the positive integers which are represented by a positive-definite integral binary quadratic form Theorem 1. Let f(X, Y) = aX ~ + b X Y + c y2 be a positive-definite integral binary quadratic form of discriminant- A ( = b 2- 4ac < 0). Let ml be the least positive in-teger represented by f. Then, for every integer n> = mi, there exist integers x and y such that n < ax 2 q- bxy + cy 2 < n + 2ml/~A1/4n ~m + m i. P r 0 0 f. Replacing the form f by an equivalent form we may suppose that m ~ = c. We define integers x and y by =F(4cn~l/2q I!4cn-Ax2) l /2-bx l x L \~) / ' Y = 2c-+1. Next we define real numbers e and 6 by e=[\T]~(4cn~l/2~j, 6 = { (4cn- Ax2)l/2-bx-}~c ' where {0} = 0- [0] denotes the fractional part of the real number 0, so tha