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Abstract. Let L = L(d) be the length of the period of the regulär continued fraction expansion of j/J where the positive integer d is not a perfect square. The first upper estimate for L was found by Lagrange and the essentially sharpest result is by Podsypanin stating that L = O (j/d log d/f2) where d/f2 is square-free. We show that for "most" /'s the stronger Statement L (d) = Oty^d) holds; more precisely, we prove the following: Let Kbe a given large number. Then there exists a natural number NQ -- NQ (K) such that for any N > N0 we have L(d) < e2K]fdfor dsatisfying N+l <d< 2N with at most cN/K2 exceptions where c denotes an absolute constant. The proof is based on Chebyshev's classical inequality in probability theory, on certain well-known relations in prime-number theory, and on Selberg's upper bound sieve. 1980 Mathematics Subject Classification (1985 Revision): 11A55, 11N35, 11K50. Let d be a positive integer that is not a perfect square and let (1) [ 0 ;* ,*2,···] = 1/5 be the regulär continued fraction expansion of j/rf. A classical result of Lagrange (see, for instance, Perron [3], p. 63) states that
Forum Mathematicum – de Gruyter
Published: Jan 1, 1990
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