Trygve Nagell

Trygve Nagell or Trygve Nagel (13 July 1895 – 24 January 1988) was a Norwegian mathematician, known for his works on Diophantine equations in number theory.^[1] He was born in Oslo and died in Uppsala.^[2]

Trygve Nagell
Trygve Nagell in the 1930s.
Born	13 July 1895 Kristiania, Norway
Died	24 January 1988(1988-01-24) (aged 92) Uppsala, Sweden
Citizenship	Norwegian
Occupation	mathematician

Education and career

He was born Nagel and adopted the spelling Nagell later in life.^[3] He received his doctorate at the University of Oslo in 1926, where his advisor was Axel Thue. He continued to lecture at the University until 1931. He was a professor at the University of Uppsala from 1931 to 1962.^[2] His doctoral students include Harald Bergström.

Contributions

Nagell proved a conjecture of Srinivasa Ramanujan that there are only five numbers that are both triangular numbers and Mersenne numbers. They are the numbers 0, 1, 3, 15, and 4095. The formula expressing the equality of a triangular number and a Mersenne number can be simplified to the equivalent form

${\displaystyle 2^{n}-7=x^{2},}$

which likewise has five solutions in natural numbers ${\displaystyle n}$ and ${\displaystyle x}$, with solutions for ${\displaystyle n\in \{3,4,5,7,15\}}$. In honor of Nagell's solution, this equation is called the Ramanujan–Nagell equation.^[4]

The Nagell–Lutz theorem is a result in the Diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It was published independently by Nagell and by Élisabeth Lutz.^[5]

In 1952, Nagell independently formulated the torsion conjecture for elliptic curves over the rationals after it was originally formulated by Beppo Levi in 1908.^[6]

Awards and honors

Nagell was appointed Commander of the Royal Norwegian Order of St. Olav in 1951, and of the Swedish Order of the Polar Star in 1952.^[1]

References

1. Ellingsrud, Geir. "Trygve Nagell". In Helle, Knut (ed.). Norsk biografisk leksikon (in Norwegian). Oslo: Kunnskapsforlaget. Retrieved 2 September 2009.
2. Henriksen, Petter, ed. (2007). "Trygve Nagell". Store norske leksikon (in Norwegian). Oslo: Kunnskapsforlaget. Retrieved 2 September 2009.
3. J. W. S. Cassels, Trygve Nagell, Acta Arithmetica, LV (1999), 108-112.
4. Turnwald, Gerhard (1990). "A note on the Ramanujan-Nagell equation". Number-theoretic analysis (Vienna, 1988–89). Lecture Notes in Math. Vol. 1452. Springer, Berlin. pp. 206–207. doi:10.1007/BFb0096992. MR 1084649.
5. Silverman, Joseph H.; Tate, John T. (2015). "2.5 The Nagell–Lutz theorem and further developments". Rational points on elliptic curves. Undergraduate Texts in Mathematics (2nd ed.). Cham: Springer. pp. 56–58. doi:10.1007/978-3-319-18588-0. ISBN 978-3-319-18587-3. MR 3363545.
6. Schappacher, Norbert; Schoof, René (1996), "Beppo Levi and the arithmetic of elliptic curves" (PDF), The Mathematical Intelligencer, 18 (1): 57–69, doi:10.1007/bf03024818, MR 1381581, S2CID 125072148, Zbl 0849.01036