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Highly composite numbers are numbers such that divisor function d(n)=sigma_0(n) (i.e., the number of divisors of n) is greater than for any smaller n. Superabundant numbers ...

The Engel expansion, also called the Egyptian product, of a positive real number x is the unique increasing sequence {a_1,a_2,...} of positive integers a_i such that ...

The base-4 method of counting in which only the digits 0, 1, 2, and 3 are used. The illustration above shows the numbers 0 to 63 represented in quaternary, and the following ...

Solve the Pell equation x^2-92y^2=1 in integers. The smallest solution is x=1151, y=120.

A quadrilateral in which a pair of opposite sides have the same length and are inclined at 60 degrees to each other (or equivalently, satisfy <A>+<B>=120 degrees). Some ...

In a given acute triangle DeltaABC, locate a point whose distances from A, B, and C have the smallest possible sum. The solution is the point from which each side subtends an ...

Given n points, find the line segments with the shortest possible total length which connect the points. The segments need not necessarily be straight from one point to ...

Synergetics coordinates are a set of triangular coordinates in their plane (or their generalization to tetrahedral coordinates in space, or the analogs in higher dimensions). ...

The triangle of numbers A_(n,k) given by A_(n,1)=A_(n,n)=1 (1) and the recurrence relation A_(n+1,k)=kA_(n,k)+(n+2-k)A_(n,k-1) (2) for k in [2,n], where A_(n,k) are shifted ...

Consider the plane figure obtained by drawing each diagonal in a regular polygon with n vertices. If each point of intersection is associated with a node and diagonals are ...