By B.M.M. de Weger

ISBN-10: 9061963753

ISBN-13: 9789061963752

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20) which explains (C). 17) we find l’ l d d’ k,k-1 k,k-1 k-1 k-2 ------------------------------ = ------------------------------ W -------------------- W -------------------- , d’ d d d’ k-1 k-1 k-2 k-1 hence l k,k-1 remains unchanged. 20), l 2 i,k Wl’ = l Wl + ( d Wd’ - l )W-------------------k-1 i,k-1 k,k-1 i,k-1 k-1 k-1 k,k-1 d k d = l Wl k,k-1 i,k-1 48 + d Wl k-2 i,k . 19) we see l’ l l l i,k i,k-1 k,k-1 i,k -------------------- = ------------------------------ - ------------------------------ W -------------------- , d d d d k k-1 k-1 k and (B) follows.

This idea has been applied in practice by Ellison b [1971 ], by Cijsouw, Korlaar and Tijdeman (appendix to Stroeker and Tijdeman [1982]), and by Hunt and van der Poorten (unpublished) for solving diophantine equations, by Steiner [1977] in connection with the Syracuse ("3WN+1") problem, and by Cherubini and Walliser [1987] (using a small home computer only) for determining all imaginary quadratic number fields with class number 1. We shall use it in Chapters 4 and 5. 3. Inhomogeneous one-dimensional approximation Davenport lemma.

In practice it appears almost always to be the case that in that situation the reduced upper bound is near to the actual largest solution, anyway so small that simple methods of finding all the solutions below that bound suffice. In the p-adic case an analogous reduction of upper bounds can be reached, 22 following a similar argument. We have for the linear form L (cf. 4)), ord (L) > c + c Wm , p 1 2 j where c , c are small constants, and m is one of the variables. 1 2 j Moreover, the variables are bounded by a large constant N , that is 0 m m+1 explicitly known.

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Algorithms for Diophantine Equations by B.M.M. de Weger

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