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A pyritohedron is an irregular dodecahedron composed of identical irregular pentagons. The name "pyritohedron" derives from that fact that a common crystal form in pyrite has ...

Find the m×n array of single digits which contains the maximum possible number of primes, where allowable primes may lie along any horizontal, vertical, or diagonal line. For ...

Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller spheres touch each other at the center of the large sphere and are tangent to ...

Given a triangle DeltaABC and the excentral triangle DeltaJ_AJ_BJ_C, define the A^'-vertex of the hexyl triangle as the point in which the perpendicular to AB through the ...

Baer's criterion, also known as Baer's test, states that a module M over a unit ring R is injective iff every module homomorphism from an ideal of R to M can be extended to a ...

A check which can be used to verify correct computations in a table of grouped classes. For example, consider the following table with specified class limits and frequencies ...

A cone with elliptical cross section. The parametric equations for an elliptic cone of height h, semimajor axis a, and semiminor axis b are x = a(h-u)/hcosv (1) y = ...

For every k>=1, let C_k be the set of composite numbers n>k such that if 1<a<n, GCD(a,n)=1 (where GCD is the greatest common divisor), then a^(n-k)=1 (mod n). Special cases ...

Given a knot diagram, it is possible to construct a collection of variables and equations, and given such a collection, a group naturally arises that is known as the group of ...

The previous prime function PP(n) gives the largest prime less than n. The function can be given explicitly as PP(n)=p_(pi(n-1)), where p_i is the ith prime and pi(n) is the ...