**Wilhelm Ljunggren**'s mother was Louise Johansen and his father was August Ljunggren who was a wheelwright. August himself had been born in Sweden into a Swedish family but had taken up permanent residence in Norway. Wilhelm attended secondary school in Oslo, graduating in 1925. Already at secondary school he had a passion for mathematics and he was an avid reader of the

*Norwegian Mathematical Journal*. (Norsk Matematisk Tidsskrift). The journal presented a collection of problems and each year the Crown Prince Olav Prize was given to the pupil who gave the best solutions to these problems. Ljunggren won the prize in his final year at secondary school.

He then entered the University of Oslo, but already he had become interested in number theory through reading papers in the *Norwegian Mathematical Journal*. Papers such as *Fermat's problem* by Øystein Ore and *On the indeterminate equation* *x*^{2} - *Dy*^{2} = 1 by Trygve Nagell were in the issue which contained the problems that he solved to win his prize and, through studying these and other papers, he was already interested in number theory before beginning his university course. Ljunggren graduated in 1931 and Thoralf Skolem, who was a student of Axel Thue, had advised him during the research for his Master's thesis. Skolem moved to the Christian Michelsen's Institute in Bergen as a Research Associate in 1930 and so Ljunggren, who wished to keep in contact with Skolem, accepted a position as a secondary school teacher in Bergen.

Ljunggren undertook research in number theory during the years he taught at the secondary school in Bergen and he submitted his doctoral thesis to the University of Oslo in 1937. Skolem worked in Bergen until 1938 when he returned to Oslo as Professor of Mathematics at the university. Ljunggren followed Skolem to Oslo, for in 1938 he became a teacher at the Hegedehaugen School. He held this position for ten years before, in 1948, he was appointed as an extraordinary professor at the University of Oslo. The University of Bergen was founded in 1946 and, three years later, Ljunggren was appointed as professor of mathematics there. In 1950 he married Else Margrethe Aas.

Although Ljunggren quickly built up a strong teaching department in Bergen he took the opportunity to return to Oslo in 1956 when he was offered the chair of mathematics there. By this time Skolem was retiring and Ljunggren took over the responsibilities of keeping Oslo as a leading institution for mathematics. He did not devote himself completely to his position at the university, however, for he had a friend Georg Schou who was trying to make a strong Technical Institute in Oslo. Ljunggren taught part-time at the Technical Institute to help his friend build its reputation.

Almost all of Ljunggren's research was on Diophantine equations. For example in *A note on simultaneous Pell equations* (1941) Ljunggren studied the simultaneous Pell equations

He proved that there is only a finite number of solutions and that it is possible to determine an upper limit for this number; in the special casex^{2}-Dy^{2}=1 andy^{2}-D_{1}z^{2}= 1.

*D*= 2,

*D*

_{1}= 3 he showed that the only solution is

*x*= 3,

*y*= 2,

*z*= 1.

One of Ljunggren's main interests was Diophantine equations of degree 4. In 1942 in the paper *Sätze über unbestimmte Gleichungen* Ⓣ he made considerable progress on problems posed by Mordell. In 1923 Mordell showed that the Diophantine equation

where the left-hand side has no squared factor inAx^{4}+Bx^{2}+C=Dy^{2},

*x*, has only a finite number of solutions. However Mordell did not find the solution, nor was he able to find bounds on the finite number of solutions. In the paper Ljunggren found bounds for the number of integer solutions for some special equations of this type.

Two other papers which he published in 1942 are *Über die Gleichung* *x*^{4} - *Dy*^{2} = 1 Ⓣ and *Zur Theorie der Gleichung* *x*^{2} +1 = *Dy*^{4} Ⓣ. In the first of these he proves that the equation in question has at most two positive integer solutions and gives an example of *D* = 1785 which does indeed have two solutions, namely *x* = 13, *y* = 4 and *x* = 239, *y* = 1352. In the second of the two papers he proves that, under certain conditions on *D*, there are again at most two positive integer solutions.

Here is one further example of the results obtained by Ljunggren. He proved that the equation

wherex^{2}-Dy^{2n}= 1

*D*+ 1 is not a square has at most two solutions if

*n*≠ 2, and if there are two solutions, these will be determined by the fundamental unit of the domain

**Z**[√

*D*] and its second or fourth power. For

*n*= 2 and 3 the result holds without condition on

*D*. In general, if

*D*+ 1 =

*u*

^{2}it holds for

*D*sufficiently large depending only on

*n*. The

*n*= 2 result had been previously proved by Nagell.

[Selberg writes [2]:-Ljunggren]was also a very gifted problem solver, contributing many original solutions to problems posed by his peers. On the other hand, Ljunggren submitted also many problems, addressed to university students as well as high-school pupils.

Wilhelm Ljunggren was an outstanding teacher at every level. He asked much of his pupils, but even more of himself. I have been told that when a school teacher, he prepared extra problems which the better pupils could try to do. He also stimulated them to further studies by helping with literature and giving advice. ... Personally I had the pleasure of collaborating with Professor Ljunggren for25years, mainly in connection with exam work. His willingness to help when we needed him, his calm and dignified manner, as well as his gentle humour, were qualities making a person one had to get fond of.

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