K Ramachandra ; S Srinivasan - A note to a paper by Ramachandra on transctndental numbers

hrj:98 - Hardy-Ramanujan Journal, January 1, 1983, Volume 6 - https://doi.org/10.46298/hrj.1983.98
A note to a paper by Ramachandra on transctndental numbers

Authors: K Ramachandra ; S Srinivasan

In this paper, we apply a combinatorial lemma to a well-known result concerning the transcendency of at least one of the numbers $\exp(\alpha_i\beta_j) (i=1, 2, 3; j=1, 2)$, where the complex numbers $\alpha_i,\beta_j$ satisfy linear independence conditions and show that for any $\alpha\neq0$ and any transcendental number $t$, we obtain that at most $\frac{1}{2}+(4N-4+\frac{1}{4})^{1/2}$ of the numbers $\exp(\alpha t^n)~(n=1,2,\ldots,N)$ are algebraic. Similar statements are given for values of the Weierstrass $\wp$-function and some connections to related results in the literature are discussed.

Volume: Volume 6
Published on: January 1, 1983
Submitted on: March 3, 2015
Keywords: algebraic numbers,Weierstrass elliptic function,transcendental numbers,[MATH] Mathematics [math]