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Baker’s Theorems and Applications

In: Pillars of Transcendental Number Theory

Author

Listed:
  • Saradha Natarajan

    (University of Mumbai, DAE Centre for Excellence in Basic Sciences)

  • Ravindranathan Thangadurai

    (Harish-Chandra Research Institute, Department of Mathematics)

Abstract

GelfondGelfond, by a refinement of his method, obtained a positive lower bound for the absolute value of $$\beta _1 \log \alpha _1+\beta _2\log \alpha _2$$ where $$\beta _1,\beta _2$$ denote algebraic numbers not both 0, and $$\alpha _1,\alpha _2$$ denote algebraic numbers not 0 or 1, with $$\log \alpha _1/\log \alpha _2$$ irrational. GelfondGelfond also remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems in number theory. In 1966–68, Baker established such a result. See his papers [1] and his prize winning book [2]. Corollaries 7.2.1, 7.2.2 and 7.2.3 resolve the multidimensional analogue of Hilbert’s seventh problem. We have chosen to give as applications, some results on Pillai’s equation, the growth of the greatest prime factor of polynomial values and effective version of Thue’ theorem; see Sects. 7.3 and 7.4.

Suggested Citation

  • Saradha Natarajan & Ravindranathan Thangadurai, 2020. "Baker’s Theorems and Applications," Springer Books, in: Pillars of Transcendental Number Theory, chapter 0, pages 107-129, Springer.
    • Handle: RePEc:spr:sprchp:978-981-15-4155-1_7
    DOI: 10.1007/978-981-15-4155-1_7
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