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Let X be a set of urelements that contains the set N of natural numbers, and let V(X) be a superstructure whose individuals are in X. Let V(^*X) be an enlargement of V(X), ...

A very general theorem that allows the number of discrete combinatorial objects of a given type to be enumerated (counted) as a function of their "order." The most common ...

The smallest composite squarefree number (2·3), and the third triangular number (3(3+1)/2). It is the also smallest perfect number, since 6=1+2+3. The number 6 arises in ...

Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique ...

A Vandermonde matrix is a type of matrix that arises in the polynomial least squares fitting, Lagrange interpolating polynomials (Hoffman and Kunze p. 114), and the ...

A set is said to be bounded from above if it has an upper bound. Consider the real numbers with their usual order. Then for any set M subset= R, the supremum supM exists (in ...

A set is said to be bounded from below if it has a lower bound. Consider the real numbers with their usual order. Then for any set M subset= R, the infimum infM exists (in R) ...

A finite, increasing sequence of integers {a_1,...,a_m} such that (a_i-1)|(a_1...a_(m-1)) for i=1, ..., m, where m|n indicates that m divides n. A Carmichael sequence has ...

If a one-parameter family of curves has index N and class M, the number tangent to a curve of order n_1 and class m_1 in general position is m_1N+n_1M.

The order-n dipole graph D_n is a multigraph consisting of two vertices and n multiple edges joining them. The dipole graph D_2 is a multigraph that can be considered to ...