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

By B.M.M. de Weger

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As far as they coincide, they coincide also with the 2 continued fraction expansion of y y , y 1 2 is needed so far that the y . If the continued fraction expansion of k th convergent with denominator known exactly, for a given (large) constant -2 as small as X . 0 X 0 , then e q > X be k 0 should be at least Most of the computer calculations done for the research on which this book reports were performed on an IBM 3083 computer at the Centraal Rekeninstituut of the University of Leiden, using the Fortran-77 language.

O | . A = | o 1 | 7 Q1 ... Qn-1 Qn where the Q space split G , of which a basis is given say, has the special form ) | | | , | 8 are large integers, that may have several hundreds of decimal i digits. We can compute a reduced basis of this lattice directly, using the 3 matrix A itself as input for the L -algorithm. But it may save time and to up the computation into several steps with increasing accuracy, as follows. , k put and Q kWl l (decimal) be a natural digits. For (j) lW(k-j)] , = [Q /10 i i Q (j) i and define J by (j+1) l (j) + J(j) .

D Wd i,k k d Wd i,j j i k=1 k-1 k j-1 j This explains the recursive formula in line (A). It remains to show that the occurring vectors c (j) i are integral. This follows from 47 j j 1 * d W S -----------------------------------Wl Wc = d W S m Wb , j d Wd i,k k j i,k k k=1 k-1 k k=1 which is integral by LLL l. p. 523, 11. (B), (C): Notice that the third and fourth line, starting from label (2), in the original algorithm, are independent of the first, second and fifth line. Thus a permutation of these lines is allowed.

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