WIlhelm Ljunggren

Wilhelm Ljunggren (1905–1973) was a Norwegian mathematician, specializing in number theory.[1][2]

Biography

Ljunggren was born October 7, 1905 in Oslo. He studied at the University of Oslo, earning a masters 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 Ph.D. from the University of Oslo in 1937. In 1938 he took a faculty position at the University of Oslo, and 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, and died January 25, 1973 in Oslo.[1]

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);[3] however, his proof was complicated, and after Louis J. Mordell conjectured that it could be simplified, simpler proofs were published by several other authors.[4][5][6]

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

Ljunggren's publications are collected in a book edited by Paulo Ribenboim.[8]

References

  1. ^ a b c 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. ^ "Wilhelm Ljunggren" (in Norwegian), Store Norske Leksikon, http://snl.no/Wilhelm_Ljunggren, retrieved 2011-12-18 .
  3. ^ Ljunggren, Wilhelm (1942), "Zur Theorie der Gleichung x2 + 1 = Dy4", Avh. Norske Vid. Akad. Oslo. I. 1942 (5): 27, MR0016375 .
  4. ^ 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, MR1092598, http://www.math.uoc.gr/~tzanakis/Papers/LjunggrenEq.pdf .
  5. ^ Draziotis, Konstantinos A. (2007), "The Ljunggren equation revisited", Colloquium Mathematicum 109 (1): 9–11, doi:10.4064/cm109-1-2, MR2308822 .
  6. ^ 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 .
  7. ^ Ljunggren, Wilhelm (1943), "Oppgave nr 2", Norsk Mat. Tidskr. 25: 29 .
  8. ^ Ribenboim, Paulo, ed. (2003), Collected papers of Wilhelm Ljunggren, Queen's papers in pure and applied mathematics, 115, Kingston, Ontario: Queen's University, ISBN 0889118361 .