Trygve Nagell or Trygve Nagel (July 13, 1895 in

Oslo Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population ...

– January 24, 1988 in

Uppsala Uppsala (, or all ending in , ; archaically spelled ''Upsala'') is the county seat of Uppsala County and the fourth-largest city in Sweden, after Stockholm, Gothenburg, and Malmö. It had 177,074 inhabitants in 2019. Located north of the c ...

) was a

Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe * Norwegians, both a nation and an ethnic group native to Norway * Demographics of Norway *The Norwegian language, including ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, known for his works on Diophantine equations in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

Education and career

He was born Nagel and adopted the spelling Nagell later in life. He received his doctorate at the

University of Oslo The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top universit ...

in 1926, where his advisor was

Axel Thue Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics. Work Thue published his first important paper in 1909. He stated in 1914 the so-calle ...

. He continued to lecture at the University until 1931. He was a professor at the

University of Uppsala Uppsala University ( sv, Uppsala universitet) is a public research university in Uppsala, Sweden. Founded in 1477, it is the oldest university in Sweden and the Nordic countries still in operation. The university rose to significance during ...

from 1931 to 1962. His doctoral students include Harald Bergström.

Contributions

Nagell proved a conjecture of Srinivasa Ramanujan that there are only five numbers that are both

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...

s and

Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...

s. 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 :

2^n-7=x^2,

which likewise has five solutions in natural numbers

n

and

x

, with solutions for

n \in\

. In honor of Nagell's solution, this equation is called the

Ramanujan–Nagell equation 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 ...

. The Nagell–Lutz theorem is a result in the

Diophantine geometry In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s, which describes

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...

points on elliptic curves over the integers. It was published independently by Nagell and by

Élisabeth Lutz Élisabeth Lutz (May 14, 1914 – July 31, 2008) 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 t ...

. In 1952, Nagell independently formulated the

torsion conjecture 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 ter ...

for elliptic curves over the rationals after it was originally formulated by

Beppo Levi Beppo Levi (14 May 1875 – 28 August 1961) was an Italian mathematician. He published high-level academic articles and books, not only on mathematics, but also on physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Aca ...

in 1908.

Awards and honors

Nagell was appointed Commander of the

Royal Norwegian Order of St. Olav Royal may refer to: People * Royal (name), a list of people with either the surname or given name * A member of a royal family Places United States * Royal, Arkansas, an unincorporated community * Royal, Illinois, a village * Royal, Iowa, a ci ...

in 1951, and of the Swedish

Order of the Polar Star The Royal Order of the Polar Star ( Swedish: ''Kungliga Nordstjärneorden'') is a Swedish order of chivalry created by King Frederick I on 23 February 1748, together with the Order of the Sword and the Order of the Seraphim. The Order of t ...

in 1952.

References

{{DEFAULTSORT:Nagell, Trygve 1895 births 1988 deaths Scientists from Oslo Norwegian mathematicians Commanders of the Order of the Polar Star University of Oslo alumni Academic staff of the University of Oslo Number theorists Royal Norwegian Society of Sciences and Letters