Computing partial information out of intractable: Powers of algebraic numbers as an example

Mika Hirvensalo*, Juhani Karhumäki, Alexander Rabinovich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)232-253
Number of pages22
JournalJournal of Number Theory
Volume130
Issue number2
DOIs
StatePublished - Feb 2010

Keywords

  • Baker theory
  • Efficient computability
  • Linear forms of logarithms

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