Search Results for ""

The twin primes constant Pi_2 (sometimes also denoted C_2) is defined by Pi_2 = product_(p>2; p prime)[1-1/((p-1)^2)] (1) = product_(p>2; p prime)(p(p-2))/((p-1)^2) (2) = ...

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, ...

An abstract manifold is a manifold in the context of an abstract space with no particular embedding, or representation in mind. It is a topological space with an atlas of ...

The boustrophedon ("ox-plowing") transform b of a sequence a is given by b_n = sum_(k=0)^(n)(n; k)a_kE_(n-k) (1) a_n = sum_(k=0)^(n)(-1)^(n-k)(n; k)b_kE_(n-k) (2) for n>=0, ...

A pair of positive integers (a_1,a_2) such that the equations a_1+a_2x=sigma(a_1)=sigma(a_2)(x+1) (1) have a positive integer solution x, where sigma(n) is the divisor ...

The canonical bundle is a holomorphic line bundle on a complex manifold which is determined by its complex structure. On a coordinate chart (z_1,...z_n), it is spanned by the ...

The parametric equations for a catenary are x = t (1) y = acosh(t/a), (2) giving the evolute as x = t-a/2sinh((2t)/a) (3) y = 2acosh(t/(2a)). (4) For t>0, the evolute has arc ...

A figurate number of the form 4n^2-3n. The first few are 1, 10, 27, 52, 85, ... (OEIS A001107). The generating function giving the decagonal numbers is ...

Given a real number x, find the powers of a base b that will shift the digits of x a number of places n to the left. This is equivalent to solving b^x=b^nx (1) or x=n+log_bx. ...

Formulas expressing trigonometric functions of an angle 2x in terms of functions of an angle x, sin(2x) = 2sinxcosx (1) cos(2x) = cos^2x-sin^2x (2) = 2cos^2x-1 (3) = ...