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The number q in a fraction p/q.

The Smarandache function mu(n) is the function first considered by Lucas (1883), Neuberg (1887), and Kempner (1918) and subsequently rediscovered by Smarandache (1980) that ...

A set S of positive integers is said to be Diophantine iff there exists a polynomial Q with integral coefficients in m>=1 indeterminates such that ...

Trigonometric functions of npi/11 for n an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 11 is not a ...

The least common multiple of two numbers a and b, variously denoted LCM(a,b) (this work; Zwillinger 1996, p. 91; Råde and Westergren 2004, p. 54), lcm(a,b) (Gellert et al. ...

A shuffle of a deck of cards obtained by successively exchanging the cards in position 1, 2, ..., n with cards in randomly chosen positions. For 4<=n<=17, the most frequent ...

A noneulerian graph is a graph that is not Eulerian. The numbers of simple noneulerian graphs on n=1, 2, ... nodes are 2, 3, 10, 30, 148, 1007, 12162, 272886, ... (OEIS ...

Closed forms are known for the sums of reciprocals of even-indexed Lucas numbers P_L^((e)) = sum_(n=1)^(infty)1/(L_(2n)) (1) = sum_(n=1)^(infty)1/(phi^(2n)+phi^(-2n)) (2) = ...

Let t(m) denote the set of the phi(m) numbers less than and relatively prime to m, where phi(n) is the totient function. Then if S_m=sum_(t(m))1/t, (1) then {S_m=0 (mod m^2) ...

Let A be the area of a simply closed lattice polygon. Let B denote the number of lattice points on the polygon edges and I the number of points in the interior of the ...