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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.

A spectrum formed by the Lagrange numbers. The only ones less than three are the Lagrange numbers, but the last gaps end at Freiman's constant. Real numbers larger than ...

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 ...

An integer which is expressible in only one way in the form x^2+Dy^2 or x^2-Dy^2 where x^2 is relatively prime to Dy^2. If the integer is expressible in more than one way, it ...