In additive number theory, Kemnitz's conjecture states that every set of lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The exact formulation of this conjecture is as follows: Let be a natural number and a set of lattice points in plane. Then there exists a subset with points such that the centroid of all points from is also a lattice point.
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| - Kemnitz-Vermutung (de)
- Conjecture de Kemnitz (fr)
- Kemnitz's conjecture (en)
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| - Die Kemnitz-Vermutung ist ein inzwischen bewiesenes Theorem der additiven Zahlentheorie. Sie besagt, dass jede Menge von Gitterpunkten in der Ebene eine große Teilmenge hat, deren Schwerpunkt wieder ein Punkt des Gitters ist. (de)
- In additive number theory, Kemnitz's conjecture states that every set of lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The exact formulation of this conjecture is as follows: Let be a natural number and a set of lattice points in plane. Then there exists a subset with points such that the centroid of all points from is also a lattice point. (en)
- La conjecture de Kemnitz est aujourd'hui un théorème de théorie additive des nombres d'après lequel, pour tout entier n > 0, parmi 4n – 3 éléments du groupe abélien fini (ℤ/nℤ)2, il en existe toujours n de somme nulle. (fr)
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| - Die Kemnitz-Vermutung ist ein inzwischen bewiesenes Theorem der additiven Zahlentheorie. Sie besagt, dass jede Menge von Gitterpunkten in der Ebene eine große Teilmenge hat, deren Schwerpunkt wieder ein Punkt des Gitters ist. (de)
- In additive number theory, Kemnitz's conjecture states that every set of lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The exact formulation of this conjecture is as follows: Let be a natural number and a set of lattice points in plane. Then there exists a subset with points such that the centroid of all points from is also a lattice point. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every integers have a subset of size whose average is an integer. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem. (en)
- La conjecture de Kemnitz est aujourd'hui un théorème de théorie additive des nombres d'après lequel, pour tout entier n > 0, parmi 4n – 3 éléments du groupe abélien fini (ℤ/nℤ)2, il en existe toujours n de somme nulle. Arnfried Kemnitz avait formulé en 1983 cette conjecture comme une généralisation du théorème d'Erdős-Ginzburg-Ziv et l'avait réduite au cas où n est premier. En 2000, (hu) l'a démontrée pour 4n – 2 éléments si n est premier et en 2001, Gao a étendu ce résultat partiel au cas où n est une puissance d'un nombre premier. La conjecture complète a été démontrée à l'automne 2003, indépendamment, par Christian Reiher (en utilisant le théorème de Chevalley-Warning) et Carlos di Fiore. (fr)
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