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The ring of integers of a number field K, denoted O_K, is the set of algebraic integers in K, which is a ring of dimension d over Z, where d is the extension degree of K over ...

For a given n, is the problem of determining if a set is mortal solvable? n=1 is solvable, n=2 is unknown, and n>=3 is unsolvable.

Let f:R->R, then the negative part of f is the function f^-:R->R defined by f^-(x)=max(-f(x),0). Note that the negative part is itself a nonnegative function. The negative ...

Let M be a finitely generated module over a commutative Noetherian ring R. Then there exists a finite set {N_i|1<=i<=l} of submodules of M such that 1. intersection ...

Place 2n balls in a bag and number them 1 to 2n, then pick half of them at random. The number of different possible sums for n=1, 2, 3, ... are then 2, 5, 10, 17, 26, ... ...

The number sign # sometimes used in mathematics to indicate the number of a quantity satisfying some condition, e.g., #{n:n>1}. The symbol is also used to denote a primorial.

The illusion illustrated above in which the bounding rectangle and inner square both appear distorted.

Through any point in the plane, there is at most one straight line parallel to a given straight line. This axiom is equivalent to the parallel postulate.

A function which has infinitely many derivatives at a point. If a function is not polygenic, it is monogenic.

A primary ideal is an ideal I such that if ab in I, then either a in I or b^m in I for some m>0. Prime ideals are always primary. A primary decomposition expresses any ideal ...