Relatively Prime
Two integers are relatively prime if they share no common factors (divisors) except 1. Using the notation 
to denotethe Greatest Common Divisor, two integers 
and 
are relatively prime if 
. Relatively prime integersare sometimes also called Strangers or Coprime and are denoted 
.
The probability that two Integers picked at random are relatively prime is 
, where
is the Riemann Zeta Function. This result is related to the fact that the Greatest Common Divisor of and , 
, can be interpreted as the number of Lattice Points in the Plane whichlie on the straight Line connecting the Vectors 
and (excluding itself). Infact, 
is the fractional number of Lattice Points Visible from theOrigin (Castellanos 1988, pp. 155-156).
Given three Integers chosen at random, the probability that no common factor will divide them all is
 | (1) |
is Apéry's Constant. This generalizes to 
random integers (Schoenfeld1976).See also Divisor, Greatest Common Divisor, Visibility
References
Castellanos, D. ``The Ubiquitous Pi.'' Math. Mag. 61, 67-98, 1988. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 3-4, 1994. Schoenfeld, L. ``Sharper Bounds for the Chebyshev Functions 
and 
, II.'' Math. Comput. 30, 337-360, 1976.
- Cascade
- Casey's Theorem
- Casimir Operator
- Cassini Ellipses
- Cassini Ovals
- Cassini Projection
- Cassini's Identity
- Cassini Surface
- Castillon's Problem
- Casting Out Nines
- Catacaustic
- Catalan Integrals
- Catalan Number
- Catalan's Conjecture
- Catalan's Constant
- Catalan's Diophantine Problem
- Catalan Solid
- Catalan's Problem
- Catalan's Surface
- Catalan's Triangle
- Catalan's Trisectrix
- Catastrophe
- Catastrophe Theory
- Categorical Game
- Categorical Variable