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A triple (a,b,c) of positive integers satisfying a<b<c is said to be geometric if ac=b^2. In particular, such a triple is geometric if its terms form a geometric sequence ...

A triple (a,b,c) of positive integers satisfying a<b<c is said to be harmonic if 1/a+1/c=2/b. In particular, such a triple is harmonic if the reciprocals of its terms form an ...

Lucas's theorem states that if n>=3 be a squarefree integer and Phi_n(z) a cyclotomic polynomial, then Phi_n(z)=U_n^2(z)-(-1)^((n-1)/2)nzV_n^2(z), (1) where U_n(z) and V_n(z) ...

It is possible to find six points in the plane, no three on a line and no four on a circle (i.e., none of which are collinear or concyclic), such that all the mutual ...

The cycloid is the locus of a point on the rim of a circle of radius a rolling along a straight line. It was studied and named by Galileo in 1599. Galileo attempted to find ...

The divergence of a vector field F, denoted div(F) or del ·F (the notation used in this work), is defined by a limit of the surface integral del ·F=lim_(V->0)(∮_SF·da)/V (1) ...

A circle is the set of points in a plane that are equidistant from a given point O. The distance r from the center is called the radius, and the point O is called the center. ...

The largest cube dividing a positive integer n. For n=1, 2, ..., the first few are 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, ... (OEIS A008834).

The representation of a number as a sum of powers of a base b, followed by expression of each of the exponents as a sum of powers of b, etc., until the process stops. For ...

A positive integer n is kth powerfree if there is no number d such that d^k|n (d^k divides n), i.e., there are no kth powers or higher in the prime factorization of n. A ...