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A note to a paper by Ramachandra on transctndental numbers

Abstract : 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.
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K Ramachandra, S Srinivasan. A note to a paper by Ramachandra on transctndental numbers. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1983, Volume 6 - 1983, pp.37 - 44. ⟨10.46298/hrj.1983.98⟩. ⟨hal-01104259⟩



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