Continued fractions constructed from prime numbers

TL;DR

This paper explores continued fractions constructed from various special prime sets, providing extensive digit values and conjecturing their transcendence, thus contributing to understanding their mathematical properties.

Contribution

It introduces continued fractions based on diverse prime sets and proposes a conjecture linking their transcendence properties.

Findings

Provided 50-digit values for continued fractions from prime-based denominators.

Identified these fractions as exceptions to Khinchin and Levy theorems.

Conjectured the transcendence of these continued fractions.

Abstract

We give 50 digits values of the simple continued fractions whose denominators are formed from a) prime numbers, b) twin primes, c) generalized $d$-twins, d) primes of the form $m^2+n^4$, e)primes of the form $m^2+1$, 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.