TY - JOUR
T1 - Computing partial information out of intractable
T2 - Powers of algebraic numbers as an example
AU - Hirvensalo, Mika
AU - Karhumäki, Juhani
AU - Rabinovich, Alexander
PY - 2010/2
Y1 - 2010/2
N2 - We consider an algorithmic problem of computing the first, i.e., the most significant digits of 2n (in base 3) and of the nth Fibonacci number. While the decidability is trivial, efficient algorithms for those problems are not immediate. We show, based on Baker's inapproximability results of transcendental numbers that both of the above problems can be solved in polynomial time with respect to the length of n. We point out that our approach works also for much more general expressions of algebraic numbers.
AB - We consider an algorithmic problem of computing the first, i.e., the most significant digits of 2n (in base 3) and of the nth Fibonacci number. While the decidability is trivial, efficient algorithms for those problems are not immediate. We show, based on Baker's inapproximability results of transcendental numbers that both of the above problems can be solved in polynomial time with respect to the length of n. We point out that our approach works also for much more general expressions of algebraic numbers.
KW - Baker theory
KW - Efficient computability
KW - Linear forms of logarithms
UR - http://www.scopus.com/inward/record.url?scp=70449527674&partnerID=8YFLogxK
U2 - 10.1016/j.jnt.2009.08.009
DO - 10.1016/j.jnt.2009.08.009
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AN - SCOPUS:70449527674
SN - 0022-314X
VL - 130
SP - 232
EP - 253
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 2
ER -