Search Results for ""

Let [a_0;a_1,a_2,...] be the simple continued fraction of a "generic" real number, where the numbers a_i are the partial quotients. Then the Khinchin (or Khintchine) harmonic ...

A rational function P(x)/Q(x) can be rewritten using what is known as partial fraction decomposition. This procedure often allows integration to be performed on each term ...

The silver ratio is the quantity defined by the continued fraction delta_S = [2,2,2,...] (1) = 2+1/(2+1/(2+1/(2+...))) (2) (Wall 1948, p. 24). It follows that ...

Let R be the set of all sets which are not members of themselves. Then R is neither a member of itself nor not a member of itself. Symbolically, let R={x:x not in x}. Then R ...

The first few numbers not known to produce palindromes when applying the 196-algorithm (i.e., a reverse-then-add sequence) are sometimes known as Lychrel numbers. This term ...

A set of real numbers x_1, ..., x_n is said to possess an integer relation if there exist integers a_i such that a_1x_1+a_2x_2+...+a_nx_n=0, with not all a_i=0. For ...

Start with an integer n, known as the digitaddition generator. Add the sum of the digitaddition generator's digits to obtain the digitaddition n^'. A number can have more ...

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given by 668+866=1534, so the next ...

A prime factorization algorithm which uses residues produced in the continued fraction of sqrt(mN) for some suitably chosen m to obtain a square number. The algorithm solves ...

int_0^z(t^mu)/(1+t)dt=z/(mu+1+((mu+1)^2z)/((mu+2)-(mu+1)z+((mu+2)^2z)/((mu+3)-(mu+2)z+...))) for mu>-1 and -1<z<=1 (Perron 1954-1957, p. 18; Borwein et al. 2004, p. 35).