Biography:Wilhelm Ljunggren
Wilhelm Ljunggren (7 October 1905 – 25 January 1973) was a Norwegian mathematician, specializing in number theory.[1]
Career
Ljunggren was born in Kristiania and finished his secondary education in 1925. He studied at the University of Oslo, earning a master's degree in 1931 under the supervision of Thoralf Skolem, and found employment as a secondary school mathematics teacher in Bergen, following Skolem who had moved in 1930 to the Chr. Michelsen Institute there. While in Bergen, Ljunggren continued his studies, earning a dr.philos. from the University of Oslo in 1937.[1][2]
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, and he also joined the Selskapet til Vitenskapenes Fremme. He was appointed as a docent at the University of Oslo in 1948, but in 1949 he returned to Bergen as a professor at the recently founded University of Bergen. He moved back to the University of Oslo again in 1956, where he served until his death in 1973 in Oslo.[1][2][3]
Research
Ljunggren's research concerned number theory, and in particular Diophantine equations.[1] He showed that Ljunggren's equation,
- X2 = 2Y4 − 1.
has only the two integer solutions (1,1) and (239,13);[4] however, his proof was complicated, and after Louis J. Mordell conjectured that it could be simplified, simpler proofs were published by several other authors.[5][6][7][8]
Ljunggren also posed the question of finding the integer solutions to the Ramanujan–Nagell equation
- 2n − 7 = x2
(or equivalently, of finding triangular Mersenne numbers) in 1943,[9] independently of Srinivasa Ramanujan who had asked the same question in 1913.
Ljunggren's publications are collected in a book edited by Paulo Ribenboim.[10]
References
- ↑ 1.0 1.1 1.2 1.3 O'Connor, John J.; Robertson, Edmund F., "Wilhelm Ljunggren", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Ljunggren.html..
- ↑ 2.0 2.1 Steenstrup, Bjørn, ed (1973). "Ljunggren, Wilhelm" (in Norwegian). Hvem er hvem?. Oslo: Aschehoug. p. 346. https://runeberg.org/hvemerhvem/1973/0346.html. Retrieved 25 April 2014.
- ↑ "Wilhelm Ljunggren" (in Norwegian). Store norske leksikon. http://www.snl.no/Wilhelm_Ljunggren. Retrieved 25 April 2014.
- ↑ Ljunggren, Wilhelm (1942), "Zur Theorie der Gleichung x2 + 1 = Dy4", Avh. Norske Vid. Akad. Oslo. I. 1942 (5): 27.
- ↑ Steiner, Ray; Tzanakis, Nikos (1991), "Simplifying the solution of Ljunggren's equation X2 + 1 = 2Y4", Journal of Number Theory 37 (2): 123–132, doi:10.1016/S0022-314X(05)80029-0, http://www.math.uoc.gr/~tzanakis/Papers/LjunggrenEq.pdf.
- ↑ Draziotis, Konstantinos A. (2007), "The Ljunggren equation revisited", Colloquium Mathematicum 109 (1): 9–11, doi:10.4064/cm109-1-2.
- ↑ Siksek, Samir (1995), Descents on Curves of Genus I, Ph.D. thesis, University of Exeter, pp. 16–17, http://www.warwick.ac.uk/~masgaj/theses/siksek_thesis.pdf.
- ↑ Cao, Zhengjun; Liu, Lihua (2017). "An Elementary Proof for Ljunggren Equation". arXiv:1705.03011 [math.NT].
- ↑ Ljunggren, Wilhelm (1943), "Oppgave nr 2", Norsk Mat. Tidsskr. 25: 29.
- ↑ Ribenboim, Paulo, ed. (2003), Collected papers of Wilhelm Ljunggren, Queen's papers in pure and applied mathematics, 115, Kingston, Ontario: Queen's University, ISBN 0-88911-836-1.
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