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The trefoil knot 3_1, also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word sigma_1^3. ...

Watson (1939) considered the following three triple integrals, I_1 = 1/(pi^3)int_0^piint_0^piint_0^pi(dudvdw)/(1-cosucosvcosw) (1) = (4[K(1/2sqrt(2))]^2)/(pi^2) (2) = ...

The j-function is the modular function defined by j(tau)=1728J(tau), (1) where tau is the half-period ratio, I[tau]>0, ...

Chess is a two-player board game believed to have been played in India as early as the sixth century AD. In different parts of this world, different chess games are played. ...

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides ...

Machin-like formulas have the form mcot^(-1)u+ncot^(-1)v=1/4kpi, (1) where u, v, and k are positive integers and m and n are nonnegative integers. Some such formulas can be ...

Riemann defined the function f(x) by f(x) = sum_(p^(nu)<=x; p prime)1/nu (1) = sum_(n=1)^(|_lgx_|)(pi(x^(1/n)))/n (2) = pi(x)+1/2pi(x^(1/2))+1/3pi(x^(1/3))+... (3) (Hardy ...

In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice ...

A ring in the mathematical sense is a set S together with two binary operators + and * (commonly interpreted as addition and multiplication, respectively) satisfying the ...

A circle is the set of points in a plane that are equidistant from a given point O. The distance r from the center is called the radius, and the point O is called the center. ...