2 6 Ja n 20 04 On the period of the continued fraction expansion of √ 2 2 n + 1 + 1
نویسنده
چکیده
It is, in general, very hard to predict the features of the continued fraction expansion of a given positive real number. If the number in question is of the form √ d, where d is a positive integer which is not a square, then its continued fraction expansion is of the form [a0, {a1, . . . , ar−1, 2a0}], where we use {. . . } to emphasize the period of the expansion. It is known that a1, . . . , ar−1 is a palindrome; i.e., ai = ar−i holds for all i = 1, . . . , r−1. The lenght r of the period is at least 1 (and this is achieved, for example, for square free numbers d of the form k2 +1 with some positive integer k), and r ≪ √ d log d (see [6]). Here, and in all what follows, we use the Vinogradov
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