**Aleksandr Osipovich Gelfond**'s father was Osip Isaacovich Gelfond who was a physician who also had an interest in philosophy. Gelfond entered Faculty of Physics and Mathematics at Moscow State University in 1924 and completed his undergraduate studies in 1927. He then began research under the supervision of Aleksandr Khinchin and Vyacheslaw Stepanov and completed his postgraduate studies in 1930.

During 1929-30 he taught mathematics at Moscow Technological College but already he had published some important papers: *The arithmetic properties of entire functions* (1929); *Transcendental numbers* (1929); and *An outline of the history and the present state of the theory of transcendental numbers* (1930). The second of these 1929 papers contained the lecture which Gelfond gave to the First All-Union Mathematics Congress held in Kharkov in 1930. These papers by Gelfond represent a major step forward in the study of transcendental numbers. The first of the papers examines the growth of an entire function which assumes integer values for integer arguments. In the second of the 1929 papers Gelfond applied this result to prove that certain numbers are transcendental, so solving a special case of Hilbert's Seventh Problem. We explain some of these ideas below.

In [5] Gelfond describes the four month visit which he made in 1930 to Germany where he spent time at both Berlin and Göttingen. He was particularly influenced by Hilbert, Siegel and Landau during his visit. After his return to Russia, Gelfond taught mathematics from 1931 at Moscow State University where he held chairs of analysis, theory of numbers and the history of mathematics. From 1933 he also worked in the Mathematical Institute of the Russian Academy of Sciences.

Gelfond developed basic techniques in the study of transcendental numbers, that is numbers that are not the solution of an algebraic equation with rational coefficients. In addition to his important work in the number theory of transcendental numbers, Gelfond made significant contributions to the theory of interpolation and the approximation of functions of a complex variable. He also contributed to the study of differential and integral equations and to the history of mathematics.

Returning to Gelfond's contributions to transcendental numbers which we mentioned above, in 1929 he conjectured that:-

If *a*_{m}, 1 ≤ *m* ≤ *n* and *b*_{m},1 ≤ *m* ≤ *n* are algebraic numbers such that { ln(*a*_{m}), 1 ≤ *m* ≤ *n* } are linearly independent over **Q**, then

b_{1}ln (a_{1}) +bln (a) + ... +b_{n}ln (a_{n}) ≠ 0.

In 1934 he proved a special case of his conjecture namely that *a*^{x} is transcendental if *a* is algebraic (*a* ≠ 0,1) and *x* is an irrational algebraic number. This result is now known as Gelfond's theorem and solved Problem 7 of the list of Hilbert problems. It was solved independently by Schneider. (In 1966 Alan Baker proved Gelfond's Conjecture in general.) Gelfond's papers in 1933 and 1934, which include his remarkable achievement, are: *Gram determinants for stationary series* (written jointly with Khinchin) (1933); *A necessary and sufficient criterion for the transcendence of a number* (1933); *Functions that take integer values at the points of a geometric progression* (1933); *On the seventh problem of D Hilbert* (1934); and *On the seventh problem of Hilbert* (1934). Gelfond addressed the Second All-Union Mathematics Congress in Leningrad in 1934) on *Transcendental numbers*.

We now look briefly at a number of books which Gelfond wrote. Some are research monographs, while others are written at undergraduate, or even high school, level. His major contributions to transcendental numbers is set out in *Transcendentnye algebraicheskie chisla* (Transcendental and algebraic numbers) (1952). In it Gelfond states that his aims are:-

.

.. to show the contemporary state of the theory of transcendental numbers, to exhibit the fundamental methods of this theory, to present the historical course of development of these methods, and to show the connections which exist between this theory and other problems in the theory of numbers.

Many of his contributions to approximation and interpolation theories are recounted in *Ischislenie konechnykh raznostey* (The calculus of finite differences) (1952). This was based on a text of the same title which Gelfond originally published in 1936. The 1936 book had been updated over the years before being rewritten for the 1952 edition. Danskin, in a review, writes:-

This book is very much in the spirit of the modern Russian school concerned with the so-called constructive theory of functions, approximative methods for the solution of differential equations, and so forth. The book is a valuable collection of results in these directions. The exposition is excellent.

Also in 1952 Gelfond published the low level *Solving equations in integers* which was translated into English in 1960. In this Gelfond states:-

This booklet is accessible to the more advanced high school students, ... , to teachers of mathematics, and to engineers.

In 1962 Gelfond published the book *Elementary methods in the analytic theory of numbers* written jointly with Linnik. Ingham writes:-

The book covers a great variety of topics in number theory, and the unifying feature is that all are treated by methods conventionally called elementary. In broad terms this means that problems are attacked by direct methods within the framework of the problems themselves, without the use of extraneous disciplines such as the theory of functions of a complex variable, Fourier analysis, trigonometric sums.

It is worth noting that although this book does not use advanced techniques, nevertheless it is not an easy book to read since the arguments are often involved and highly complex.

A further text by Gelfond is *Residues and their applications* (1966). The chapter titles of this book are: Residues; Singular points and series representations of a function; Expansion of a function in a series and properties of the gamma function; Some functional identities and asymptotic estimates; and Laplace transformation and some problems which are solved by the use of residue theory.

The authors of [4] (in the translation [5]) tell us something about Gelfond as a mathematician:-

His mathematical ability was esteemed, above all, for its originality. Many eminent mathematicians think roughly along the same lines as less eminent ones, though more rapidly and in a more organised way; Gelfond always thought in his own way, one that was unconventional and quite original. For this reason, his outstanding work was for a long time in subjects in which there had been intensive research.

The authors of [4] also write about Gelfond as a teacher of mathematics:-

Gelfond devoted much time and effort to training of young scholars, with tact, kindness, and a sincere sensitivity, being remarkably individualistic himself, he valued and respected individuality in his pupils. Without restricting their views and tastes, he knew how to pass on to them his own devotion to science.

As to his interests outside cutting edge mathematical research [5]:-

... he was an expert(of professional standard or almost professional standard)at chess, literature, mineralogy and the history of science. He was an exceptionally good companion, and easily won the friendship and confidence of people of very different kinds.

**Article by:** *J J O'Connor* and *E F Robertson*

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