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Teichmüller's theorem asserts the existence and uniqueness of the extremal quasiconformal map between two compact Riemann surfaces of the same genus modulo an equivalence ...

The abc conjecture is a conjecture due to Oesterlé and Masser in 1985. It states that, for any infinitesimal epsilon>0, there exists a constant C_epsilon such that for any ...

For a triangle DeltaABC and three points A^', B^', and C^', one on each of its sides, the three Miquel circles are the circles passing through each polygon vertex and its ...

The constant s_0 in Schnirelmann's theorem such that every integer >1 is a sum of at most s_0 primes. Of course, by Vinogradov's theorem, it is known that 4 primes suffice ...

The geometric span is largest possible distance between two points drawn from a finite set of points. It is therefore closely related to the generalized diameter of a closed ...

Let p be an odd prime, k be an integer such that pk and 1<=k<=2(p+1), and N=2kp+1. Then the following are equivalent 1. N is prime. 2. There exists an a such that ...

Let s_i be the orders of singular points on a curve (Coolidge 1959, p. 56). Harnack's first theorem states that a real irreducible curve of order n cannot have more than ...

Vizing's theorem states that a graph can be edge-colored in either Delta or Delta+1 colors, where Delta is the maximum vertex degree of the graph. A graph with edge chromatic ...

Given a property P, if P(x)∼x as x->infty (so, using asymptotic notation, the number of numbers less than x not satisfying the property P is o(x), where o(x) is one of the ...

A point B is said to lie between points A and C (where A, B, and C are distinct collinear points) if AB+BC=AC. A number of Euclid's proofs depend on the idea of betweenness ...