The Journal of the Indian Mathematical Society

T. N. Shorey ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Colaba, Bombay 400 005, IN

Abstract

Ifav α₁, α₂, β₁ are rational numbers satisfying (i) α₁ > 0, α₂ > 0 are multiplicatively independent (ii) the size of α₁, α₂, β₁, respectively, do not exceed S₁, S₁ and (log S₁)¹⁰⁰ (100 is quite unimportant), then | β₁ log α₁ - log α₂| > C(∈) exp ( - (log S₁)^2+z) (1) where ∈ > 0 is an arbitrary fixed constant and C(∈) is an effectively computable positive constant depending only on ∈.

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T. N. Shorey ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Colaba, Bombay 400 005, IN

Abstract

The purpose of this note is to prove the following:

THEOREM 1. If α₁, α₂, α₃, β₁, β₂ are rational numbers satisfying (i) α₁ > 0, α₂ > 0, α₃ > 0 are multiplicatively independent (ii) the size of α_i, ≤ S₁, i = 1, 2, 3, and that of β_i, ≤ (log S₁)¹⁰⁰ = S, i = 1, 2, (100 is quite unimportant), then

| β₁ log α₁ + β₂ log α₂ - log α₃ | > C(∈) exp ( - (log ₁)^8+∈)

where ∈ > 0 is an arbitrary fixed constant and C(∈) is an effectively computable positive constant depending only on ∈.

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T. N. Shorey ¹

Affiliations
1 Tata Institute of Fundamental Research, IN

Abstract

Let r and s be integers satisfying s≠0 and r²+4s≠0. Let u₀, u₁ be integers and we define
u_m=ru_m-1+su_m-2, m=2,3,... (1).

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