Trygve Nagell

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Trygve Nagell in the 1930s. TrygveNagellOB.F06294c.jpg
Trygve Nagell in the 1930s.

Trygve Nagell (July 13, 1895 in Oslo January 24, 1988 in Uppsala) was a Norwegian mathematician, known for his works on Diophantine equations in number theory. [1]


Education and career

He received his doctorate at the University of Oslo in 1926, and lectured 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.


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

which likewise has five solutions in natural numbers and , with solutions for . In honor of Nagell's solution, this equation is called the Ramanujan–Nagell equation. [3]

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. [4]

In 1952, Nagell independently formulated the torsion conjecture for elliptic curves over the rationals after it was originally formulated by Beppo Levi in 1908. [5]

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]

Related Research Articles

Diophantine equation Polynomial equation whose integer solutions are sought

In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.

Elliptic curve Algebraic curve

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which, after a linear change of variables, consists of solutions (x,y) for:

Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu. The integers 23 and 32 are two powers of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive powers. That is to say, that

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In mathematics, 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 is named for Trygve Nagell and Élisabeth Lutz.

In mathematics, Ribet's theorem is a statement in number theory concerning properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof of the epsilon conjecture was a significant step towards the proof of Fermat's Last Theorem. As shown by Serre and Ribet, the Taniyama–Shimura conjecture and the epsilon conjecture together imply that Fermat's Last Theorem is true.

Arithmetic geometry Branch of algebraic geometry focused on problems in number theory

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Congruent number

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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 solved in integers where one of the variables appears as an exponent. It is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture.

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In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.

Élisabeth Lutz was a French mathematician. The Nagell–Lutz theorem in Diophantine geometry describes the torsion points of elliptic curves; it is named after Lutz and Trygve Nagell, who both published it in the 1930s.

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In mathematics, the Mordell–Weil theorem states that for an abelian variety over a number field , the group of K-rational points of is a finitely-generated abelian group, called the Mordell–Weil group. The case with an elliptic curve and the rational number field Q is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.

Wilhelm Ljunggren Norwegian mathematician

Wilhelm Ljunggren was a Norwegian mathematician, specializing in number theory.

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. The torsion conjecture has been completely resolved in the case of elliptic curves.


  1. 1 2 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. Turnwald, Gerhard (1990). "A note on the Ramanujan-Nagell equation". Number-theoretic analysis (Vienna, 1988–89). Lecture Notes in Math. 1452. Springer, Berlin. pp. 206–207. doi:10.1007/BFb0096992. MR   1084649.
  4. 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.
  5. 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