Linear Forms in the Logarithms of Algebraic Numbers with Small Coefficients II

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Authors

  • T. N. Shorey
     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 αi, ≤ S1, i = 1, 2, 3, and that of βi, ≤ (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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Published

1974-12-01

How to Cite

Shorey, T. N. (1974). Linear Forms in the Logarithms of Algebraic Numbers with Small Coefficients II. The Journal of the Indian Mathematical Society, 38(1-4), 285–292. Retrieved from https://www.informaticsjournals.com/index.php/jims/article/view/16702

Issue

Volume 38, Issue 1-4, March-December 1974

Section

Articles

License

Copyright (c) 1974 T. N. Shorey

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

 

References

A. BAKER, Linear forms in the logarithms of algebraic numbers, Mathematika, 13 (1966), pp 204-16.

A. BAKER, Linear forms in the logarithms of algebraic numbers, Mathematika, 14 (1967), pp 102-07.

K. RAMACHANDRA, A note on numbers with a large prime factor III, Acta Arithmetical 19(1967), pp 49-62.

K. RAMACHANDRA, Lectures on Transcendental Numbers, The Ramanujan Institute, Madras (1969).

T.N. SHORE Y, Linear forms in the logarithms of algebraic numbers with small coefficients I Jour. Indian Math. Soc. 38 (1974), 271-284.

R. TIJDEMAN, An Auxiliary Result in the theory of Transcendental Numbers Journal of Number Theory 5 (1973), 80-94.