Trygve Nagell is a famous Scandinavian mathematician who was born in Oslo, 1895. He studied at the University of Oslo and was in particular influenced by Axel Thue. He was Professor in Uppsala from 1931 until his retirement in 1962. He died in 1988. Much of his works are devoted to Number Theory, especially Diophantine equations. He wrote 138 papers and four books. Among his papers about 70 are written in French, 50 in German, 20 in English and the others in Scandinavian languages. His most famous book is Introduction to Number Theory (Wiley 1951; Zbl 0042.26702), reprinted by Chelsea (1964). His small book L’Analyse Indéterminée de Degré supérieur, 63 p. (1929; JFM 55.0712.02), is reproduced in the present collection.
The four volumes of this collection contain the complete collection of Nagell’s papers, that is more than 2000 pages. A study of Nagell’s work published by Cassels in Acta Arith. 55, 109-112 (1990) is reproduced in the beginning of the first volume. I am unable to improve on this presentation, the reader is warmly invited to consult it.
The publication of this collection is a big event for people interested in Elementary Number Theory and Diophantine Equations. Nagell’s papers are scattered in many places, some appear in rare booklets, and were very often almost impossible to find (even on the net). I am quite surprised that it was possible to realize this collection and Paulo Ribenboim did an extraordinary job to succeed in this challenge, the mathematical community is greatly indebted to him.
It is a great pleasure to read these papers which are elegant, written very clearly and which contain very efficient methods and beautiful results. I quote just three of Nagell’s results.
– For example, concerning Catalan’s equation Nagell proved already very precise results from 1921 to 1934 in the cases \(X^2-Y^p=1\) and \(X^3-Y^p=1\) (he solved completely this second case, the first one was solved by Ko Chao in 1965).
– He has been the first, in 1928, to show that if a curve of genus one contains a rational point \(P\), then this curve is birationally equivalent over the rational to some curve \(\Gamma\) defined by the equation \(X^3-AX-B=Y^2\), where \(A\) and \(B\) are rational integers and \(P\) transforms into the point at infinity of \(\Gamma\) and, in 1935 he showed that the rational points \((a,b)\) of \(\Gamma\) which are of finite order satisfy \(a\), \(b\in {\mathbb Z}\) and either \(b=0\) or \(b^2\) divides the discriminant \(4A^3-27B^2\) (in 1937, E. Lutz gave a \(p\)-adic proof of this result which is now known as the Nagell-Lutz theorem).
– He was also the first to solve completely Ramanujan’s Diophantine equation \(x^2+7=2^n\), using a \(p\)-adic method in 1948, but his result remained almost ignored for more than ten years and an independent solution was published in 1959 by Chowla, Lewis and Skolem.