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By Richard Bellman

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We denote our field by Q(Jd). If IX = X + yJd with x, y E Q, then we define IX' = X - yJd, [IV, §1] QUADRATIC NUMBERS AND PERIODICITY 51 and call IX' the conjugate of IX. The reader will easily verify by direct computation that for IX, PE Q(Jd} we have (IX + py = IX' + P' and We define the trace of IX to be Tr(lX) = IX + IX', and its norm to be Both the trace and norm are rational numbers (obvious). We note that is a root of the quadratic polynomial (X - IX}(X - The quadratic equation for IX IX') = X2 - Tr(IX}X IX + N(IX}.

In this book, we are principally interested in specific numbers, and we shall omit the proof of Theorem 7, but give a partial result (Corollary 3 of Theorem 8 below) consistent with our point of view. g. higher dimensional ones, and also has a very good error term. This is important, because in dealing with specific numbers, the expression of the error term reflects the special nature of the number under consideration in an essential way. For further work on this, cf. also Gallagher [9]. It is a problem to determine specific numbers, and functions t/J for which A.

II, §3. ASYMPTOTIC APPROXIMATIONS Throughout this section, we let such that t/I be a positive function 00 L t/I(q) q=l ~ 1, decreasing, 26 ASYMPTOTIC APPROXIMATIONS diverges. We let 'I'(N) = IN [II, §3] t/J(t) dt. : ",(N) be the number of solutions in integers q, p of the inequalities o < qIX - P < t/J(q) 1 ~ q < N. and To simplify the notation, we shall omit the + sign, and also we usually omit the indices IX, t/J on A.. , but such an estimate was proved only recently for almost all numbers. We shall state this result (without proof).

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