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The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral ...

A regular polygon is an n-sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular ...

A repunit prime is a repunit (i.e., a number consisting of copies of the single digit 1) that is also a prime number. The base-10 repunit (possibly probable) primes ...

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 ...

There exist infinitely many odd integers k such that k·2^n-1 is composite for every n>=1. Numbers k with this property are called Riesel numbers, while analogous numbers with ...

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 ...

The Rogers-Ramanujan continued fraction is a generalized continued fraction defined by R(q)=(q^(1/5))/(1+q/(1+(q^2)/(1+(q^3)/(1+...)))) (1) (Rogers 1894, Ramanujan 1957, ...

For |q|<1, the Rogers-Ramanujan identities are given by (Hardy 1999, pp. 13 and 90), sum_(n=0)^(infty)(q^(n^2))/((q)_n) = 1/(product_(n=1)^(infty)(1-q^(5n-4))(1-q^(5n-1))) ...

A run is a sequence of more than one consecutive identical outcomes, also known as a clump. Let R_p(r,n) be the probability that a run of r or more consecutive heads appears ...

A series is an infinite ordered set of terms combined together by the addition operator. The term "infinite series" is sometimes used to emphasize the fact that series ...