By B.M.M. de Weger

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Extra resources for Algorithms for Diophantine Equations

Sample text

A ----------------------------------------------------------------- |eiWx-1| = 2W|sin(1Wx)| . and that 2Wsin(1Wx)/x is a 2 2 positive and even function, that decreases on 0 < x < a . Hence it takes its Proof. Note that minimal value at ----- ----- x = a . The first inequality now follows. The second one p can be proved in a similar way. 3. p-adic numbers and functions. In this section we mention the facts about p-adic numbers and functions that we use. For details we refer to Bachman [1964] and Koblitz [1977], [1980].

2) leads to a large range of convergents of y for which the values of it appears to be the case that NqWjN qWj p/q are all extremely small. 5] ). This method has been used in practice by Baker and Davenport [1969] as we already mentioned, by Ellison, Ellison, Pesek, Stahl and Stall [1972], by Steiner [1986], and by Gasl [1988]. We shall use it in Chapter 4. 8 for the multi-dimensional inhomogeneous case, can be used in the one-dimensional case b as well, as has been demonstrated in de Weger [1989 ].

We have the following result. 10) we denote the distance to the nearest integer). 2) satisfy 1 ( 2 ) . X < -----Wlog q Wc/|y |WX d 9 2 00 Proof. 10) we infer 2WX /q < NqW(j-x Wy+x )+x W(qWy-p)N < qW|L/y | + |x |/q . 11). 10) is not true for the first convergent with denominator one should try some further convergents. 11) yields a reduced upper bound for X of size log X , 0 0 as desired. 10) (a situation which is very unlikely to occur, as experiments show), then not all is lost, since then only very few exceptional possible solutions have to be checked.