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An amicable pair (m,n) consists of two integers m,n for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other. Amicable ...

If k|n, then the complete k-uniform hypergraph on n vertices decomposes into 1-factors, where a 1-factor is a set of n/k pairwise disjoint k-sets. Brouwer and Schrijver ...

The Barth sextic is a sextic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points, namely 65. The surface was ...

A bicubic graph is a bipartite cubic graph. Tutte (1971) conjectured that all 3-connected bicubic graphs are Hamiltonian (the Tutte conjecture), but a number of bicubic ...

Roman (1984, p. 26) defines "the" binomial identity as the equation p_n(x+y)=sum_(k=0)^n(n; k)p_k(y)p_(n-k)(x). (1) Iff the sequence p_n(x) satisfies this identity for all y ...

The nth cabtaxi number is the smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes. The name derived from the taxicab ...

"Casting out nines" is an elementary check of a multiplication which makes use of the congruence 10^n=1 (mod 9). Let decimal numbers be written a=a_n...a_2a_1a_0, ...

(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+alpha^2y=0 (1) for |x|<1. The Chebyshev differential equation has regular singular points at -1, 1, and infty. It can be solved by series ...

Select three points at random on the circumference of a unit circle and find the distribution of areas of the resulting triangles determined by these three points. The first ...

A circumhyperbola is a circumconic that is a hyperbola. A rectangular circumhyperbola always passes through the orthocenter H and has center on the nine-point circle ...