## Abstract

Alan Baker, who died on the 4th of February of this year, was born in England on the 19th of August 1939. In 1965 he defended his doctoral dissertation titled ‘Some Aspects of Diophantine Approximation’ at Trinity College Cambridge under the guidance of Harold Davenport. It is a very unusual fact that eight of his papers, including [1], which is discussed in the next section, had appeared in print before he submitted his doctoral dissertation. Baker was awarded the Fields Medal in 1970 at the International Congress of Mathematicians at Nice. Other honors he received are detailed in the Article-in-a-Box in this issue.

In the present text, we will discuss his paper [1] and the ‘Baker theory of linear forms in logarithms’, which started with the series of four papers [2, 3, 4, 5] published in the British journal *Mathematika*. This new theory was an impressive breakthrough in the field of Diophantine approximation and we will briefly present some of its applications. For precise references to original papers, including those mentioned in the next sections, the reader is directed to the monographs [6, 7, 8, 9].

## Keywords

Baker’s theory Thue equation Diophantine equations irrationality measure transcendence## Preview

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## Suggested Reading

- [1]A Baker, Rational Approximations to xxxx and Other Algebraic Numbers,
*Quart. J. Math.Oxford Ser.*, Vol.15, pp.375–383, 1964.CrossRefGoogle Scholar - [2]A Baker, Linear Forms in the Logarithms of Algebraic Numbers. I,
*Mathematika*, Vol.12, pp.204–216, 1966.CrossRefGoogle Scholar - [3]A Baker, Linear Forms in the Logarithms of Algebraic Numbers. II,
*Mathematika*, Vol.14, pp.102–107, 1967.CrossRefGoogle Scholar - [4]A Baker, Linear Forms in the Logarithms of Algebraic Numbers. III,
*Mathematika*, Vol.14, pp.220–228, 1967.CrossRefGoogle Scholar - [5]A Baker, Linear Forms in the Logarithms of Algebraic Numbers. IV,
*Mathematika*, Vol.15, pp.204–216, 1968.CrossRefGoogle Scholar - [6]A Baker and G Wüstholz, Logarithmic Forms and Diophantine Geometry,
*New Mathematical Monographs*, 9, Cambridge University Press, Cambridge, 2007.Google Scholar - [7]Y Bugeaud, Linear Forms in Logarithms and Applications,
*IRMA Lectures in Mathematics and Theoretical Physics*, 28, Zürich: European Mathematical Society, 2018.Google Scholar - [8]T N Shorey and R Tijdeman, Exponential Diophantine equations,
*Cambridge Tracts in Mathematics*, Vol.87, Cambridge University Press, Cambridge, 1986.Google Scholar - [9]M Waldschmidt, Diophantine Approximation on Linear Algebraic Groups, Transcendence Properties of the Exponential Function in Several Variables,
*Grundlehren Math. Wiss.*, Vol.326, Springer, Berlin, 2000.Google Scholar - [10]R Tijdeman, On the Equation of Catalan,
*Acta Arith.*, Vol.29, pp.197–209, 1976.CrossRefGoogle Scholar - [11]P Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan’s Conjecture,
*J. reine angew. Math.*, 572, pp.167–195, 2004.Google Scholar