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Erdős proved that there exist at least one prime of the form 4k+1 and at least one prime of the form 4k+3 between n and 2n for all n>6.

For every positive integer n, there exists a circle which contains exactly n lattice points in its interior. H. Steinhaus proved that for every positive integer n, there ...

A prime factorization algorithm.

A closed interval is an interval that includes all of its limit points. If the endpoints of the interval are finite numbers a and b, then the interval {x:a<=x<=b} is denoted ...

In a cochain complex of modules ...->C^(i-1)->^(d^(i-1))C^i->^(d^i)C^(i+1)->..., the module B^i of i-coboundaries is the image of d^(i-1). It is a submodule of C^i and is ...

Two elements x and y of a set S are said to be commutative under a binary operation * if they satisfy x*y=y*x. (1) Real numbers are commutative under addition x+y=y+x (2) and ...

A function whose range is in the complex numbers is said to be a complex function, or a complex-valued function.

A complex vector space is a vector space whose field of scalars is the complex numbers. A linear transformation between complex vector spaces is given by a matrix with ...

Compass and straightedge geometric constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, ...

A continuous map is a continuous function between two topological spaces. In some fields of mathematics, the term "function" is reserved for functions which are into the real ...