## Relatively Prime

Two integers are relatively prime if they share no common factors (divisors) except 1. Using the notation to denote the Greatest Common Divisor, two integers and are relatively prime if . Relatively prime integers are 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 which lie on the straight Line connecting the Vectors and (excluding itself). In fact, is the fractional number of Lattice Points Visible from the Origin (Castellanos 1988, pp. 155-156).

Given three Integers chosen at random, the probability that no common factor will divide them all is

 (1)

where is Apéry's Constant. This generalizes to random integers (Schoenfeld 1976).

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.