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Finding the densest not necessarily periodic sphere packing.

An a×b rectangle can be packed with 1×n strips iff n|a or n|b.

The polyhedron resulting from letting each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces.

Let each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces. Call the resulting polyhedron the local cell. Then the local density is ...

The fraction eta of a volume filled by a given collection of solids.

An integer m such that if p|m, then p^2|m, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, ...

A Ruth-Aaron pair is a pair of consecutive numbers (n,n+1) such that the sums of the prime factors of n and n+1 are equal. They are so named because they were inspired by the ...

Maximize the amount of floor space which can be covered with a fixed tile (Hoffman 1998, p. 173).

A parallelogram (parallelepiped) containing the minimum repeatable elements of a circle (sphere) packing.

A set of three conjectures proposed by Weil in 1942 for extending Riemann hypothesis-like statements to classes of mathematical structures. The conjectures were proved by ...