Lemma of Thue

The lemma Thue , with some authors set of Thue called, is a theorem of elementary number theory , a sub-region of mathematics . It goes back to the Norwegian mathematician Axel Thue and plays a role in investigations into Diophantine equations . The proof is based on the Dirichlet drawer principle .

Formulation of the lemma

Thue's lemma can be summarized as follows:

If there is an integer and a positive natural number that is relatively prime to it, there is always a pair of positive natural numbers, which on the one hand correspond to the inequalities ${\displaystyle l}$ ${\displaystyle m}$ ${\displaystyle (x,y)}$

(U)   ${\displaystyle x,y<{\sqrt {m}}}$

is sufficient and, on the other hand, at least one of the two congruence relations

(K ₁ )   ${\displaystyle l\cdot y\equiv x{\pmod {m}}}$

or.

(K ₂ )   ${\displaystyle l\cdot y\equiv -x{\pmod {m}}}$

Fulfills.

In particular:

For an integer and a prime number , which does not divide , one always finds a pair of integers which correspond to the inequalities ${\displaystyle l}$ ${\displaystyle m}$ ${\displaystyle l}$ ${\displaystyle (x,y)}$

(U ^* )   ${\displaystyle 0<|x|,|y|<{\sqrt {m}}}$

is sufficient and at the same time the congruence relation

(K ^* )   ${\displaystyle l\cdot y\equiv x{\pmod {m}}}$

Fulfills.

In addition, the following applies even more generally:

Let be whole numbers and be there and coprime and at the same time satisfy the inequalities . ${\displaystyle l,m,u,v}$ ${\displaystyle l}$ ${\displaystyle m}$ ${\displaystyle 0<u,v\leq m<u\cdot v}$

Then there is a pair of integers which satisfies the inequalities and and at the same time satisfies one of the two above congruence relations K _i . ${\displaystyle (x,y)}$ ${\displaystyle 0<x<u}$ ${\displaystyle 0<y<v}$

Following sentence

With the Thueschen Lemma (and with the help of the First Supplementary Theorem to the law of reciprocity ) a well-known theorem about the representability of certain prime numbers as sums of squares can be proven, which was first proved by Leonhard Euler (however, Albert Girard and Pierre de Fermat are also said to have been known ):

A prime number which satisfies the congruence relation always has a sum representation and this representation is, apart from the order of the two summands, even unambiguous. ${\displaystyle p}$ ${\displaystyle p\equiv 1{\pmod {4}}}$ ${\displaystyle p=a^{2}+b^{2}\;(a,b\in \mathbb {N} )}$

Historical note

Axel Thues Lemma goes back to one of his works from 1915. As early as 1913, a mathematician named L. Aubry had presented a related result. Various authors have subsequently made a number of generalizations about both.

literature

• Alfred Brauer , RL Reynolds: On a theorem of Aubry-Thue . In: Canadian Journal of Mathematics . tape 3 , 1951, pp. 367-374 ( MR0048487 ).
• Peter Bundschuh : Introduction to number theory (=  Springer textbook ). 6th, revised and updated edition. Springer Verlag, Berlin / Heidelberg 2008, ISBN 978-3-540-76490-8 .
• Siegfried Gottwald , Hans-Joachim Ilgauds , Karl-Heinz Schlote (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch, Thun 1990, ISBN 3-8171-1164-9 . MR1089881
• Hartmut Menzer: Number Theory . Five selected topics in number theory. Oldenbourg Verlag, Munich 2010, ISBN 978-3-486-59674-8 .
• Trygve Nagell , Atle Selberg , Sigmund Selberg, Knut Thalberg (Eds.): Selected Mathematical Papers of Axel Thue . With an introduction by Carl Ludwig Siegel . Universitetsforlaget, Oslo / Bergen / Tromsø 1977, ISBN 82-00-01649-8 ( MR1567083 ).
• Kenneth H. Rosen (Ed.): Handbook of Discrete and Combinatorial Mathematics (=  Discrete Mathematics and its Applications ). CRC Press, 2000, ISBN 0-8493-0149-1 .
• Wacław Sierpiński : Elementary Theory of Numbers (=  North-Holland Mathematical Library . Volume 31 ). 2nd revised and expanded edition. North-Holland ( inter alia), Amsterdam ( inter alia ) 1988, ISBN 0-444-86662-0 ( MR0930670 ).
• A. Scholz , B. Schoeneberg (Ed.): Introduction to number theory (=  Göschen Collection . Volume 1131 ). Walter de Gruyter, Berlin 1966 ( MR0071438 ).

Individual evidence and notes

1. ↑ Peter Bundschuh: Introduction to Number Theory. 2008, p. 154 ff
2. ↑ ^{a b} Wacław Sierpiński: Elementary Theory of Numbers 1988, pp. 30-31
3. ↑ ^{a b} Hartmut Menzer: Number theory. 2010, p. 273 ff.
4. ↑ ^{a b c} Kenneth H. Rosen (Ed.): Handbook of Discrete and Combinatorial Mathematics. 2000, p. 234
5. ↑ Bundschuh, op.cit., P. 155
6. ↑ Menzer, op.cit., P. 274
7. ^ A. Scholz, B. Schoeneberg: Introduction to number theory. 1966, p. 44 ff
8. ↑ This somewhat more general version of the lemma goes back to the introduction to number theory by Scholz and Schoeneberg (see p. 44).
9. ↑ Bundschuh, op. Cit., Pp. 154–156
10. ↑ See the discussion of the work by Brauer and Reynolds in the Mathematical Reviews (MR0048487). Regarding the first occurrence of the lemma, the corresponding article Thue's lemma in the English language Wikipedia refers to a work submitted by Thue in 1902. See: Trygve Nagell et al. (Ed.): Selected Mathematical Papers of Axel Thue. 1977, pp. 57-75, pp. 539-549!
11. ^ Alfred Brauer, RL Reynolds: On a theorem of Aubry-Thue. In: Canadian J. Math. , 3, p. 367 ff.