Wilhelm Ljunggren

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Wilhelm Ljunggren (7 October 1905 – 25 January 1973) was a Norwegian mathematician, specializing in number theory.[1]


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]


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]

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,[8] independently of Srinivasa Ramanujan who had asked the same question in 1913.

Ljunggren's publications are collected in a book edited by Paulo Ribenboim.[9]


  1. ^ a b c d O'Connor, John J.; Robertson, Edmund F., "Wilhelm Ljunggren", MacTutor History of Mathematics archive, University of St Andrews ..
  2. ^ a b Steenstrup, Bjørn, ed. (1973). "Ljunggren, Wilhelm". Hvem er hvem? (in Norwegian). Oslo: Aschehoug. p. 346. Retrieved 25 April 2014. 
  3. ^ "Wilhelm Ljunggren". Store norske leksikon (in Norwegian). Retrieved 25 April 2014. 
  4. ^ Ljunggren, Wilhelm (1942), "Zur Theorie der Gleichung x2 + 1 = Dy4", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27, MR 0016375 .
  5. ^ Steiner, Ray; Tzanakis, Nikos (1991), "Simplifying the solution of Ljunggren's equation X2 + 1 = 2Y4" (PDF), Journal of Number Theory, 37 (2): 123–132, doi:10.1016/S0022-314X(05)80029-0, MR 1092598 .
  6. ^ Draziotis, Konstantinos A. (2007), "The Ljunggren equation revisited", Colloquium Mathematicum, 109 (1): 9–11, doi:10.4064/cm109-1-2, MR 2308822 .
  7. ^ Siksek, Samir (1995), Descents on Curves of Genus I (PDF), Ph.D. thesis, University of Exeter, pp. 16–17 .
  8. ^ Ljunggren, Wilhelm (1943), "Oppgave nr 2", Norsk Mat. Tidsskr., 25: 29 .
  9. ^ 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 .