By B.M.M. de Weger
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22) X = X . 21) then fails, 1 0 C . 22) yields a reduced lower bound for 0 Proof. , x 1 n the lattice point ~ L of size 0 < X < X 1 log X 0 . 1) with X as above. Then n-1 2 2 ~2 2 2 ~2 = g W S x + L < (n-1)Wg WX + L , i 1 i=1 2 |x| and ~ |L-gWCWL| < which is < nWX 1 n n S |x |W|[gWCWy ]-gWCWy | < S |x | , i i i i i=1 i=1 . 22) follows at once. 4. The parameter g is used to keep the "rounding-off error" 54 |[gWCWy ]-gWCWy | i i relatively small. This is of importance only if C is not very large, usually only if one wants to make a further reduction step after the first step has already been made.
P and also it may be useful to compute it by 1+x ( 3 5 ) . = 2W x + x /3 + x /5 + ... p 1-x 9 0 log If --------------- x # 0 (mod p) there exists a log p and k e N x = x # 1 (mod p) then log x can be computed, since p k such that x _ 1 (mod p) , and then 1 Wlogp(91+(xk-1))0 k ----- and the above given Taylor series can be used to compute log x . Note that p in computing the above mentioned Taylor series there will be factors p in the denominators of the terms. Hence, to find the first m p-adic digits of log (1+x) , it is not enough to compute only the first m/ord (x) terms of p p the Taylor series, but the first k terms must be taken into account, where k is the smallest integer satisfying kWord (x) - log k/log p > m .
Let In K K with given norm. , a be elements of O that are Q-linearly independent. 1 D K Then ZWa * ... * ZWa is called an order of K if it is a subring of the 1 D ’maximal order’ O . K any algebraic integer can be written as a product of irreducible elements. Here an irreducible element (prime element) is an element that has no integral divisors but its own associates. However, this decomposition into primes need not be unique. Ideals can also be decomposed into prime ideals, and this decomposition is unique.
Algorithms for Diophantine Equations by B.M.M. de Weger