By B.M.M. de Weger

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Extra resources for Algorithms for Diophantine Equations

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5_(Schinzel). let D Let p be the discriminant of be elements of K , where L = log max be prime. Let D K = Q(rD) . Let x’, x", c’, c" be a squarefree integer, and x = x"/x’ and c = c"/c’ are algebraic integers. Put ( |eWD|1/4, Nx’Wc’N, Nx’Wc"N, Nx"Wc’N, Nx"Wc"N ) , 9 0 denotes the maximal absolute value of the conjugates of g e K . r Let p be a prime ideal of K with norm Np = p . Put j = 2/rWlog p , n m v = ord (p) . If x or c is a p-adic unit and x $ c , then where NgN p n m 6 7 -2 4 4Wr+4 ( ord (x -c ) < 10 Wj Wv WL Wp W log max(|m|,|n|)+vWLWpr+2/L)3 .

So it has proved to be useful. However, we prefer another method, for several reasons. Firstly, it is close to the homogeneous case as described in the previous section, whereas the generalized Davenport method has no obvious counterpart for the homogeneous solutions for which the linear form under the condition y case. L Secondly, it actually produces is almost as near to zero as possible X < X . Specifically, if a linear relation between the 0 exists, but had not been noticed before (a situation that may occur in i practice, cf.

Or the more rapidly converging series 1+x ( 3 5 ) . = 2W x + x /3 + x /5 + ... 1-x 9 0 log --------------- |x| For very small this method works fast, whereas for larger |x| the following idea works well. 00000001 , say, and store them. Now compute 1 such that , which is a matter of a few divisions of a multi-precision number with a rational number with small numerator and denominator (11 and 10) only, that can be done fast. 00000001 2 3 x x 3 e N0 such that . 1 . 3 2 1 When computing all this, one should take care of having the rounding off errors at each addition/multiplication under control.

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