The Product eπe \pi Is Irrational

This note shows that the product $e \pi$ of the natural base $e$ and the circle number $\pi$ is an irrational number.Comment: Twelve Pages. Improved Version. keywords: Irrational number; Natural base, Circle number; Unbounded partial quotients. arXiv admin note: text overlap with arXiv:1212.408

Explicit irrationality measures for continued fractions

AbstractLet τ=[a0;a1,a2,…], a0∈N, an∈Z+, n∈Z+, be a simple continued fraction determined by an infinite integer sequence (an). We are interested in finding an effective irrationality measure as explicit as possible for the irrational number τ. In particular, our interest is focused on sequences (an) with an upper bound at most (ank), where a>1 and k>0. In addition to our main target, arithmetic of continued fractions, we shall pay special attention to studying the nature of the inverse function z(y) of y(z)=zlogz