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Taylor's inequality is an estimate result for the value of the remainder term R_n(x) in any n-term finite Taylor series approximation. Indeed, if f is any function which ...

For a subgroup H of a group G, the index of H, denoted (G:H), is the cardinal number of the set of left cosets of H in G (which is equal to the cardinal number of the set of ...

Every odd integer n is a prime or the sum of three primes. This problem is closely related to Vinogradov's theorem.

There are a number of algebraic identities involving sets of four vectors. An identity known as Lagrange's identity is given by (AxB)·(CxD)=(A·C)(B·D)-(A·D)(B·C) (1) ...

A theorem due to Conway et al. (1997) which states that, if a positive definite quadratic form with integer matrix entries represents all natural numbers up to 15, then it ...

If alpha is any number and m and n are integers, then there is a rational number m/n for which |alpha-m/n|<=1/n. (1) If alpha is irrational and k is any whole number, there ...

A spectrum sequence is a sequence formed by successive multiples of a real number a rounded down to the nearest integer s_n=|_na_|. If a is irrational, the spectrum is called ...

A parallelogram polyomino is a polyomino such that the intersection with every line perpendicular to the main diagonal is a connected segment. The number of parallelogram ...

Find a square number x^2 such that, when a given integer h is added or subtracted, new square numbers are obtained so that x^2+h=a^2 (1) and x^2-h=b^2. (2) This problem was ...

A piecewise linear function is a function composed of some number of linear segments defined over an equal number of intervals, usually of equal size. For example, consider ...