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Two numbers are homogeneous if they have identical prime factors. An example of a homogeneous pair is (6, 72), both of which share prime factors 2 and 3: 6 = 2·3 (1) 72 = ...

The Machin-like formula 1/4pi=2cot^(-1)(3)+cot^(-1)(7). The other two-term Machin-like formulas are Euler's Machin-like formula, Hermann's formula, and Machin's formula.

A fraction p/q>=1. A fraction with p/q<1 is called a proper fraction. Therefore, the special cases 1/1, 2/2, 3/3, etc. are generally considered to be improper.

n divides a^n-a for all integers a iff n is squarefree and (p-1)|(n-1) for all prime divisors p of n. Carmichael numbers satisfy this criterion.

The continued fraction ((x+1)^n-(x-1)^n)/((x+1)^n+(x-1)^n)=n/(x+)(n^2-1)/(3x+)(n^2-2^2)/(5x+...).

A sequence defined from a finite sequence a_0, a_1, ..., a_n by defining a_(n+1)=max_(i)(a_i+a_(n-i)).

A maximal sum-free set is a set {a_1,a_2,...,a_n} of distinct natural numbers such that a maximum l of them satisfy a_(i_j)+a_(i_k)!=a_m, for 1<=j<k<=l, 1<=m<=n.

A sequence defined from a finite sequence a_0, a_1, ..., a_n by defining a_(n+1)=mex_(i)(a_i+a_(n-i)), where mex is the mex (minimum excluded value).

Let q be a positive integer, then Gamma_0(q) is defined as the set of all matrices [a b; c d] in the modular group Gamma Gamma with c=0 (mod q). Gamma_0(q) is a subgroup of ...

The set lambda of linear Möbius transformations w which satisfy w(t)=(at+b)/(ct+d), where a and d are odd and b and c are even. lambda is a subgroup of the modular group ...