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Perfect numbers are positive integers n such that n=s(n), (1) where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), or equivalently ...

If there is an integer 0<x<p such that x^2=q (mod p), (1) i.e., the congruence (1) has a solution, then q is said to be a quadratic residue (mod p). Note that the trivial ...

Let N steps of equal length be taken along a line. Let p be the probability of taking a step to the right, q the probability of taking a step to the left, n_1 the number of ...

The regular icosahedron (often simply called "the" icosahedron) is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron ...

The value for zeta(2)=sum_(k=1)^infty1/(k^2) (1) can be found using a number of different techniques (Apostol 1983, Choe 1987, Giesy 1972, Holme 1970, Kimble 1987, Knopp and ...

The 120-cell is a finite regular four-dimensional polytope with Schläfli symbol {5,3,3}. It is also known as the hyperdodecahedron or hecatonicosachoron, and is composed of ...

An abundant number, sometimes also called an excessive number, is a positive integer n for which s(n)=sigma(n)-n>n, (1) where sigma(n) is the divisor function and s(n) is the ...

There are four completely different definitions of the so-called Apollonius circles: 1. The set of all points whose distances from two fixed points are in a constant ratio ...

A 4-cusped hypocycloid which is sometimes also called a tetracuspid, cubocycloid, or paracycle. The parametric equations of the astroid can be obtained by plugging in n=a/b=4 ...

The nth coefficient in the power series of a univalent function should be no greater than n. In other words, if f(z)=a_0+a_1z+a_2z^2+...+a_nz^n+... is a conformal mapping of ...