Axel Thue Über die Auflösbarkeit einiger unbestimmten Gleichungen

Abstract

Axel Thue was born in 1863 in Tønsberg, a town in the south-east of Norway. He studied science and mathematics at the University of Oslo, and mathematics during stays in Leipzig and Berlin. He held a position as senior teacher in mechanics at an engineering college in Trondheim from 1894 to 1903. In 1903 he was appointed professor of applied mathematics (which meant rational mechanics) at the University of Oslo, a position he held until his death in 1922.

Thue never held a position with teaching obligations in mathematics, but he published a number of papers in his special fields of research. These were mainly n number theory, but also in mathematical logic, geometry and mechanics. He was an extremely original thinker, and very rarely referred to results of other researchers. His results in number theory spanned from simple rest theorems for congruence to deep results on Diophantine equations and rational approximation of algebraic numbers. He proved the existence of transcendental numbers as well as later the transcendence of the numbers e and π , but he did not publish these results since they were already known. His most famous result is a theorem on Diophantine equations and in that connection on rational approximation of irrational algebraic numbers. The theorem on Diophantine equations states that such an equation of the form F(x , y) = C, where C is an integer and F( x , y ) is a homogenous and irreducible polynomial of degree n ≥ 3, has only a finite number of solutions. The approximation result states that for every irrational algebraic number α of degree n ≥ 3 and every positive number ε , the inequality (..., see the pdf) has only a finite number of solutions p/q with p and q integers.

Thue published two papers in the DKNVS Skrifter. One of these was in a problem in geometry, the other on a topic in number theory. The first part of this last paper contains conditions for solvability of a certain type of Diophantine equations. The second part deals with a problem related to «Fermat’s last theorem». Thue here shows by an elementary but complicated argument of «descent» that the equation (..., see the pdf) has no integer solutions with P and Q relatively prime and R not divisible by 3. A consequence is that the equation (..., see the pdf) has no integer solutions where z is not divisible by 3.