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 -