By B.M.M. de Weger

**Read Online or Download Algorithms for Diophantine Equations PDF**

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**Extra resources for Algorithms for Diophantine Equations**

**Example text**

Put 1 n Let W p , for a fixed prime n S x Wy . i i i=1 L = b + We classify such linear forms according to three criteria: -----L homogeneous if b = 0 , inhomogeneous if -----L one-dimensional if -----L real if y i e R n = 2 , for all The reason that the case b $ 0 ; multi-dimensional if i , p-adic if n = 2 y i e W p n > 3 ; for all i . is called one-dimensional is that in the homogeneous case the linear form L = x Wy + x Wy 1 1 2 2 leads to studying the simple, one-dimensional continued fraction expansion of -y /y .

4 below), but too recently to be taken into account here. First we deal with real/complex linear forms in logarithms. We quote the result of Waldschmidt [1980]. 4_(Waldschmidt). , b e Z ( n > 2 ) . Let 1 n 1 n positive real numbers satisfying 1/D < V < ... < V and 1 n V j where > max (9 h(aj), |log aj|/D log a ) 0 for [K:Q] = D . , n . , n is an arbitrary but fixed determination of j + the logarithm of a . Let V = max(V ,1) for j = n, n-1 , and put j j j 29 n S bjWlog aj . L = Put B = j=1 max |b | .

2. 7). 1) , then 1 1 ( ) 1 X < -----Wlog cW(A+2)/|y | + -----Wlog X . d 9 2 0 d Remark. 1 is sharp for large X Proof. b follows. We can apply Lemma only. 5) yield (a 2 -1 > q W|y |/|L| > q W|y |Wc Wexp(dWX) . 1(i). In practice it does not often occur that A p is large. Therefore this lemma is useful indeed. Summarizing, this case comes down to computing the continued fraction of a real number to a certain precision, and establishing that it has no extremely large partial quotients. 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[1971 ], by Cijsouw, Korlaar and Tijdeman (appendix to Stroeker and Tijdeman [1982]), and by Hunt and van der Poorten (unpublished) for solving diophantine equations, by Steiner [1977] in connection with the Syracuse ("3WN+1") problem, and by Cherubini and Walliser [1987] (using a small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.