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Consider the inequality sigma(n)<e^gammanlnlnn for integer n>1, where sigma(n) is the divisor function and gamma is the Euler-Mascheroni constant. This holds for 7, 11, 13, ...

Let sum_(n=1)^(infty)u_n(x) be a series of functions all defined for a set E of values of x. If there is a convergent series of constants sum_(n=1)^inftyM_n, such that ...

Every polynomial equation having complex coefficients and degree >=1 has at least one complex root. This theorem was first proven by Gauss. It is equivalent to the statement ...

The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not ...

Mills (1947) proved the existence of a real constant A such that |_A^(3^n)_| (1) is prime for all integers n>=1, where |_x_| is the floor function. Mills (1947) did not, ...

Lagrange's continued fraction theorem, proved by Lagrange in 1770, states that any positive quadratic surd sqrt(a) has a regular continued fraction which is periodic after ...

König's line coloring theorem states that the edge chromatic number of any bipartite graph equals its maximum vertex degree. In other words, every bipartite graph is a class ...

Several flavors of the open mapping theorem state: 1. A continuous surjective linear mapping between Banach spaces is an open map. 2. A nonconstant analytic function on a ...

Gödel's first incompleteness theorem states that all consistent axiomatic formulations of number theory which include Peano arithmetic include undecidable propositions ...

A general reciprocity theorem for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. If R is a number field and R^' a finite ...