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

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Authors

  • T. N. Shorey
     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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Published

1974-12-01

How to Cite

Shorey, T. N. (1974). Linear Forms in the Logarithms of Algebraic Numbers with Small Coefficients I. The Journal of the Indian Mathematical Society, 38(1-4), 271–284. Retrieved from https://www.informaticsjournals.com/index.php/jims/article/view/16701

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 I, Mathematika, 13(1966), pp. 204-16.

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

P. ERDOS, On consecutive Integers, Nieuw, Arch. Voor. Wiskunde, 3 (1951), pp. 268-273.

K. RAMACHANDRA, A note on numbers with a large prime factor III, Acta Arithmetica, 19 (1971).

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

K. RAMACHANDRA, Contributions to the Theory of Transcendental Numbers II, Acta Arithmetica, Vol. 14 (1968).

T.N. SHOREY, On a theorem of Ramachandra Acta Airth. 20 (1972), 215-221.

R. TuDEMAN, On the maximal distance of numbers with a large prime factor Jour. Lond. Math. Soc. 5 (1972), 313-320.

R. TIJDEMAN, An auxliary Result in the theory of transcendental Numbers Jour. Num. Theory. 5 (1973), 80-94.