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Let (L,<=) be any complete lattice. Suppose f:L->L is monotone increasing (or isotone), i.e., for all x,y in L, x<=y implies f(x)<=f(y). Then the set of all fixed points of f ...

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The nth homotopy group of a topological space X is ...

Let K be a number field with r_1 real embeddings and 2r_2 imaginary embeddings and let r=r_1+r_2-1. Then the multiplicative group of units U_K of K has the form ...

The smallest positive composite number and the first even perfect square. Four is the smallest even number appearing in a Pythagorean triple: 3, 4, 5. In the numerology of ...

The AC method is an algorithm for factoring quadratic polynomials of the form p(x)=Ax^2+Bx+C with integer coefficients. As its name suggests, the crux of the algorithm is to ...

A lucky number of Euler is a number p such that the prime-generating polynomial n^2-n+p is prime for n=1, 2, ..., p-1. Such numbers are related to the imaginary quadratic ...

The sequence defined by e_0=2 and the quadratic recurrence equation e_n=1+product_(i=0)^(n-1)e_i=e_(n-1)^2-e_(n-1)+1. (1) This sequence arises in Euclid's proof that there ...

A general quadratic Diophantine equation in two variables x and y is given by ax^2+cy^2=k, (1) where a, c, and k are specified (positive or negative) integers and x and y are ...

The number theta in the quadric (x^2)/(a^2+theta)+(y^2)/(b^2+theta)+(z^2)/(c^2+theta)=1 is called the parameter.

An m-ary n-ic polynomial (i.e., a homogeneous polynomial with constant coefficients of degree n in m independent variables).