Abouabdillah's theorem
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Abouabdillah's theorem refers to two distinct theorems in mathematics: one in geometry and one in number theory.
[edit] Geometry
In geometry, similarities of an Euclidean space preserve circles and spheres. Conversely, the Abouabdillah's theorem states that every injective or surjective transformation of an Euclidean space that preserves circles or spheres is a similarity.
More precisely:
Theorem. - Let E be an Euclidean affine space of dimension ≥2. Then.
1°) Every surjective application f:E→E that transforms any four concyclic points into four concyclic points is a similarity.
2°) Every injective application f:E→E that transforms any circle into a circle is a similarity.
Proof. - Cf. [1].
[edit] Number Theory
In Number Theory the Abouabdillah's theorem is about antichains of N. ( An antichain of N, for divisibility, is a set of non null integers such that no one is divisible by another. It possible to prove using Dilworth's theorem that the maximal cardinality of an antichain of E2n = {1,2,...,2n} is n).
Abouabdillah's Theorem. Let c∈E2n , c = 2kc', c' odd. Then E2n contains an antichain of cardinality n containing c if and only if 2n < 3k+1c'.
