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Selected mathematical papers. With a biography written by Viggo Brun, and an introduction by Carl Ludwig Siegel. Edited by Trygve Nagell, Atle Selberg, Sigmund Selberg, Knut Thalberg. (English) Zbl 0359.01021

Oslo-Bergen-Tromsø: Universitetsforlaget. LVIV, 592 p. N.Kr. 200.00; $ 40.00 (1977).
Axel Thue (1863–1922) who taught applied mathematics at Trondheim (1894–1903) and Oslo (1892–94, 1903–22), wrote almost fifty papers in number theory, logic, geometry and mechanics. His major contribution was in the first of these, particularly in the areas of Diophantine equations and rational approximation of algebraic numbers. In this volume are facsimile copies of 35 papers written in German and Norwegian covering his research in number theory and logic. Fourteen papers in other areas, not reproduced, are listed. Brief English summaries are provided for eight of the nine Norwegian papers.
Thue’s most influential work is illustrated by the following two theorems, proved in 1908 and 1909 respectively: (1) Let \(F\) be an irreducible polynomial of degree exceeding 2 with integer coefficients and \(c\) be an arbitrary integer; then there are at most finitely many integer pairs \((p,q)\) for which \(q^rF(p/q)=c\) holds. (2) Let \(p\) be a positive algebraic number of degree \(n\), and let \(c\) and \(k\) be any positive numbers; then the inequality \(| qp- p| <cq^{-(\frac 12 n+ k)}\) is satisfied for at most finitely many positive integer pairs \((p,q)\). (Pages 219 and 232.)
The latter theorem was extended by Siegel in 1921 and Roth (for a Fields Medal) in 1958. A modern account of the relation between these theorems and of subsequent investigation can be found in K. B. Stolarsky [Algebraic numbers and diophantine approximation (1974; Zbl 0285.10022)]. Fermat’s conjecture seems to have held Thue’s attention during his whole career, and he published several notes on it at various times. One note gives a combinatorial demonstration of Fermat’s Little Theorem.
The collection is preceded by a biography of Thue written by Viggo Brun and two articles on Thue’s number-theoretic work by Carl Ludwig Siegel. The second of these, originally published in [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1970, 169–195 (1970; Zbl 0215.34601)], analyzes in detail nine of Thue’s published papers. (On page 589, ”Nr. 33” should read ”Nr. 32”.)

MSC:

01A75 Collected or selected works; reprintings or translations of classics
01A05 General histories, source books
01A60 History of mathematics in the 20th century
11-03 History of number theory
11J68 Approximation to algebraic numbers
11D59 Thue-Mahler equations

Biographic References:

Thue, Axel