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Distinguished mathematician who won the Fields medal for his contribution to number theory
In 1966 the start of a new era in number theory was marked by Alan Baker, who has died aged 78, joining the department of pure mathematics at Cambridge University. With a cascade of papers, he had published solutions to a series of problems from a line of inquiry that went back to the third-century mathematician Diophantus of Alexandria. On the basis of this exceptional work, in 1970 Alan was awarded the Fields medal, one of the discipline's highest distinctions.
The interest of Diophantus's approach to equations lies in whether they can be solved in ways that produce only whole numbers, or integers. From school, we know Pythagoras's theorem for right-angled triangles: if the sides are 3, 4 and 5 units long, then ), and there are other whole-number solutions, or Pythagorean triples, that can be found with squared numbers (5,12,13; 7, 24, 25; and infinitely many more). But can the equivalent be done with cubed numbers, or numbers at higher powers?
The underlying diophantine equation is . This has been a subject of particular fascination since 1637, when the Frenchman Pierre de Fermat noted in the margin of his translation of Diophantus that whole-number solutions for it could not be found beyond squaring, and that he had "discovered a truly marvellous demonstration of this proposition that this margin is too narrow to contain".
Alan's way into this question lay in one of the most difficult areas of mathematics, transcendental number theory. It searches for and investigates the properties of non-algebraic numbers, a classification that includes , 3.1415... , the ratio of the circumference to the diameter of a circle, and Euler's number, , 2.7182..., important in the study of logarithms.
During a golden period in the 1930s and 40s, the Russian mathematician AO Gelfond and the German PhD student Theodor Schneider had given a solution (1934), to the seventh of the 23 unsolved problems described in 1900 by the German mathematician David Hilbert, and showed it to be transcendental. Hilbert regarded this seventh problem as much more difficult to prove than the Riemann hypothesis, also on his list, which is now one of the seven millennium problems posed by the Clay Mathematics Institute; the reward offered for a solution to any of these is $1m, as they would play a tremendous role in the future development of mathematics.
Gelfond and Schneider pushed forward the techniques they had developed into different directions but essentially it seemed that the limits of their method had been reached. So it was a cause of much surprise that Alan got into this very difficult part of number theory from around 1964 onwards, and worked on it in Cambridge on his own.
Alan was born in London. His father, Barney, had also been a brilliant mathematician in his youth, but had had to earn a living rather than pursue his studies. Barney married Betty Sohn and they set up home in Forest Gate, east London. From a very early age, their son showed signs of mathematical brilliance, which they encouraged.
From Stratford grammar school, Alan gained a scholarship to study at University College London. He graduated with a first and went to Trinity College, Cambridge, to study for an MA and PhD (1965) with Harold Davenport, one of the leading number theorists at the time, with many international connections.
Alan was awarded the Fields medal for research into what are known as linear forms in logarithms. It had been recognised for a long time that the solution to a number of outstanding problems in number theory would depend on this work, including Carl Friedrich Gauss's class number problem, published in his textbook Disquisitiones Arithmeticae (1798). The Gelfond-Schneider method could not be applied to deal with the general linear forms in logarithms that were needed for these applications; Alan managed to find a spectacular new approach.
Using his theory of logarithmic forms he was able to settle the Gauss conjecture, and also to find all solutions in integers of the diophantine equations of the type , known as Mordell equations, after the 20th-century British mathematician Louis Mordell. Alan's work deduced that there are only a finite number of integer solutions of the equation with z a fixed given integer and which can be computed.
This fundamental insight has been developed spectacularly by others. Gerd Faltings, another Fields medallist, showed in 1983 that diophantine equations of this type including the Fermat equation have only finitely many solutions, a result conjectured by Mordell. However, the solutions cannot be computed explicitly. Andrew Wiles finally published a proof (1995) that for integer solutions with no common factor the Fermat equation has only the trivial solutions 0, 1 and -1 for powers higher than squaring, as stated by Fermat.
Alan's insights have also found applications in many fields of mathematics quite remote from number theory. From 1974 until 2006 he was professor of pure mathematics at Cambridge University, then made emeritus.
He was a modest and frugal person who had continued to live at Trinity College from his days as a graduate student. From the mid-90s he was proud of occupying the college's best flat, next to the Master's Lodge, which even had a separate, almost secret staircase down to the fellows'garden, which he could overlook from his living room. There he loved to play bowls with friends.
During an international conference in Zurich organised by the mathematical research institute FIM in 1999 on the occasion of a his 60th birthday, he gave a speech in which he expressed regret at never having married.
Travelling around the world, in particular to Hong Kong, was his pleasure, as well as photography and theatre. He often visited the US, in particular the Institute for Advanced Study in Princeton. Among the many honours he received were the Adams prize, election to the Royal Society and the Academia Europaea, and to honorary fellowship of UCL.
Alan Baker, mathematician, born 19 August 1939; died 4 February 2018
Gisbert Wüstholz
9 April 2018 © Guardian Newspapers Limited