English

Continued fractions constructed from prime numbers

Number Theory 2010-09-28 v2 History and Overview

Abstract

We give 50 digits values of the simple continued fractions whose denominators are formed from a) prime numbers, b) twin primes, c) generalized dd-twins, d) primes of the form m2+n4m^2+n^4, e)primes of the form m2+1m^2+1, f) Mersenne primes and g) primorial primes. All these continued fractions belong to the set of measure zero of exceptions to the theorems of Khinchin and Levy. We claim that all these continued fractions are transcendental numbers. Next we propose the conjecture which indicates the way to deduce the transcendence of some continued fractions from transcendence of another ones.

Keywords

Cite

@article{arxiv.1003.4015,
  title  = {Continued fractions constructed from prime numbers},
  author = {Marek Wolf},
  journal= {arXiv preprint arXiv:1003.4015},
  year   = {2010}
}

Comments

Considerably extended version of previous submission. We add the discussion of transcendentality of continued fractions constructed from prime numbers. 35 pages and 9 Figures

R2 v1 2026-06-21T15:00:25.933Z