Wilhelm Ljunggren (7 October 1905 – 25 January 1973) was a

Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe *Norwegians, both a nation and an ethnic group native to Norway *Demographics of Norway *The Norwegian language, including the ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, specializing in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

Career

Ljunggren was born in

Kristiania Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population of ...

and

finished Finished may refer to: * ''Finished'' (novel), a 1917 novel by H. Rider Haggard * ''Finished'' (film), a 1923 British silent romance film * "Finished" (short story), a science fiction short story by L. Sprague de Camp See also *Finishing (disa ...

his secondary education in 1925. He studied at the

University of Oslo The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top universit ...

, earning a

master's degree A master's degree (from Latin ) is an academic degree awarded by universities or colleges upon completion of a course of study demonstrating mastery or a high-order overview of a specific field of study or area of professional practice.

in 1931 under the supervision of

Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem ...

, and found employment as a secondary school mathematics teacher in

Bergen Bergen (), historically Bjørgvin, is a city and municipality in Vestland county on the west coast of Norway. , its population is roughly 285,900. Bergen is the second-largest city in Norway. The municipality covers and is on the peninsula of ...

, following Skolem who had moved in 1930 to the

Chr. Michelsen Institute The Chr. Michelsens Institutt for Videnskap og Åndsfrihet (CMI) was founded in 1930, and is currently the largest centre for development research in Scandinavia. In 1992, the Department for Natural Science and Technology established the ''Ch ...

there. While in Bergen, Ljunggren continued his studies, earning a dr.philos. from the University of Oslo in 1937. In 1938 he moved to work as a teacher at Hegdehaugen in Oslo. In 1943 he became a fellow of the

Norwegian Academy of Science and Letters The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick Univer ...

, and he also joined the Selskapet til Vitenskapenes Fremme. He was appointed as a

docent The title of docent is conferred by some European universities to denote a specific academic appointment within a set structure of academic ranks at or below the full professor rank, similar to a British readership, a French " ''maître de conf ...

at the University of Oslo in 1948, but in 1949 he returned to Bergen as a professor at the recently founded

University of Bergen The University of Bergen ( no, Universitetet i Bergen, ) is a research-intensive state university located in Bergen, Norway. As of 2019, the university has over 4,000 employees and 18,000 students. It was established by an act of parliament in 194 ...

. He moved back to the University of Oslo again in 1956, where he served until his death in 1973 in Oslo.

Research

Ljunggren's research concerned number theory, and in particular

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

s. He showed that Ljunggren's equation, :''X''² = 2''Y''⁴ − 1. has only the two integer solutions (1,1) and (239,13); however, his proof was complicated, and after Louis J. Mordell conjectured that it could be simplified, simpler proofs were published by several other authors. Ljunggren also posed the question of finding the integer solutions to the

Ramanujan–Nagell equation In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be so ...

:2^''n'' − 7 = ''x''² (or equivalently, of finding

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinea ...

Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...

s) in 1943, independently of

Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...

who had asked the same question in 1913. Ljunggren's publications are collected in a book edited by

Paulo Ribenboim Paulo Ribenboim (born March 13, 1928) is a Brazilian-Canadian mathematician who specializes in number theory. Biography Ribenboim was born into a Jewish family in Recife, Brazil. He received his BSc in mathematics from the University of São Pa ...

References

{{DEFAULTSORT:Ljunggren, Wilhelm 1905 births 1973 deaths Number theorists University of Oslo alumni University of Bergen faculty University of Oslo faculty Norwegian schoolteachers Members of the Norwegian Academy of Science and Letters 20th-century Norwegian mathematicians