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Two integers are relatively prime if they share no common positive 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 plot above plots and along the two
axes and colors a square black if
and white otherwise (left figure) and simply colored according to
(right figure).
Wolfram Web Resources »
Two numbers can be tested to see if they are relatively prime in Mathematica using CoprimeQ[m, n].
13,542 entries
Last updated: Wed Feb 4 2015
Two distinct primes
and ,
Created, developed, and
nurtured by Eric Weisstein
at Wolfram Research
Relative primality is not transitive. For example,
and
.
are always relatively prime,
The probability that two integers
and
, as are any positive integer powers of distinct primes
and
, but
.
picked at random are relatively prime is
(1)
(OEIS A059956; Cesàro and Sylvester 1883; Lehmer 1900; Sylvester 1909; Nymann 1972; Wells 1986, p. 28; Borwein
and Bailey 2003, p. 139; Havil 2003, pp. 40 and 65; Moree 2005), 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
(2)
(OEIS A088453; Wells 1986, p. 29), where
is Apéry's constant (Wells 1986, p. 29). In general, the probability that
random numbers lack a th power common divisor is
(Cohen 1959, Salamin 1972, Nymann 1975,
Schoenfeld 1976, Porubský 1981, Chidambaraswamy and Sitaramachandra Rao 1987, Hafner et al. 1993).
Interestingly, the probability that two Gaussian integers
and
are relatively prime is
(3)
(OEIS A088454), where
is Catalan's constant (Pegg; Collins and Johnson 1989; Finch 2003, p. 601).
Similarly, the probability that two random Eisenstein integers are relatively prime is
(4)
(OEIS A088467), where
(5)
(Finch 2003, p. 601), which can be written analytically as
(6)
(7)
(OEIS A086724), where
is the trigamma function
Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as
the asymptotic densities of squarefree integers of these types.
SEE ALSO:
Divisor, Greatest Common Divisor, Hafner-Sarnak-McCurley Constant, Squarefree, Visible Point
REFERENCES:
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2/4/15, 1:38 PM
Relatively Prime -- from Wolfram MathWorld
http://mathworld.wolfram.com/RelativelyPrime.html
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.
Castellanos, D. "The Ubiquitous Pi." Math. Mag. 61, 67-98, 1988.
Chidambaraswamy, J. and Sitaramachandra Rao, R. "On the Probability That the Values of M Polynomials Have a Given G.C.D." J.
Number Th. 26, 237-245, 1987.
Cohen, E. "Arithmetical Functions Associated with Arbitrary Sets of Integers." Acta Arith. 5, 407-415, 1959.
Collins, G. E. and Johnson, J. R. "The Probability of Relative Primality of Gaussian Integers." Proc. 1988 Internat. Sympos.
Symbolic and Algebraic Computation (ISAAC), Rome (Ed. P. Gianni). New York: Springer-Verlag, pp. 252-258, 1989.
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 3-4, 1994.
Hafner, J. L.; Sarnak, P.; and McCurley, K. "Relatively Prime Values of Polynomials." In A Tribute to Emil Grosswald: Number
Theory and Related Analysis (Ed. M. Knopp and M. Seingorn). Providence, RI: Amer. Math. Soc., 1993.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York:
Hyperion, pp. 38-39, 1998.
Lehmer, D. N. "An Asymptotic Evaluation of Certain Totient Sums." Amer. J. Math. 22, 293-355, 1900.
Moree, P. "Counting Carefree Couples." 30 Sep 2005. http://arxiv.org/abs/math.NT/0510003.
Nagell, T. "Relatively Prime Numbers. Euler's
-Function." §8 in Introduction to Number Theory. New York: Wiley, pp. 23-26, 1951.
Nymann, J. E. "On the Probability That
Positive Integers Are Relatively Prime." J. Number Th. 4, 469-473, 1972.
Nymann, J. E. "On the Probability That
Positive Integers Are Relatively Prime. II." J. Number Th. 7, 406-412, 1975.
Pegg, E. Jr. "The Neglected Gaussian Integers." http://www.mathpuzzle.com/Gaussians.html.
Porubský, S. "On the Probability That K Generalized Integers Are Relatively H-Prime." Colloq. Math. 45, 91-99, 1981.
Salamin, E. Item 53 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence
Laboratory, Memo AIM-239, p. 22, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item53.
Schoenfeld, L. "Sharper Bounds for the Chebyshev Functions
and
, II." Math. Comput. 30, 337-360, 1976.
Sloane, N. J. A. Sequences A059956, A086724, A088453, A088454, and A088467 in "The On-Line Encyclopedia of Integer
Sequences."
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 28-29, 1986.
Referenced on Wolfram|Alpha: Relatively Prime
CITE THIS AS:
Weisstein, Eric W. "Relatively Prime." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com
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