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Szemerédi's theorem states that every sequence of integers that has positive upper Banach density contains arbitrarily long arithmetic progressions. A corollary states that, ...

The smallest positive composite number and the first even perfect square. Four is the smallest even number appearing in a Pythagorean triple: 3, 4, 5. In the numerology of ...

The third prime number, which is also the second Fermat prime, the third Sophie Germain prime, and Fibonacci number F_4. It is an Eisenstein prime, but not a Gaussian prime, ...

An atlas is a collection of consistent coordinate charts on a manifold, where "consistent" most commonly means that the transition functions of the charts are smooth. As the ...

There are two curves known as the butterfly curve. The first is the sextic plane curve given by the implicit equation y^6=x^2-x^6 (1) (Cundy and Rollett 1989, p. 72; left ...

Given a chord PQ of a circle, draw any other two chords AB and CD passing through its midpoint. Call the points where AD and BC meet PQ X and Y. Then M is also the midpoint ...

Combinatorial geometry is a blending of principles from the areas of combinatorics and geometry. It deals with combinations and arrangements of geometric objects and with ...

A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2={0,1} has a representation phi ...

A non-Euclidean geometry, also called Lobachevsky-Bolyai-Gauss geometry, having constant sectional curvature -1. This geometry satisfies all of Euclid's postulates except the ...

A set-theoretic term having a number of different meanings. Fraenkel (1953, p. 37) used the term as a synonym for "finite set." However, according to Russell's definition ...