*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let be a random number from written as a simple Continued Fraction

(1) |

(2) |

Gauß (1800) showed that if is the probability that , then

(3) |

(4) |

**References**

Babenko, K. I. ``On a Problem of Gauss.'' *Soviet Math. Dokl.* **19**, 136-140, 1978.

Daudé, H.; Flajolet, P.; and Vallée, B. ``An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction.'' Submitted.

Durner, A. ``On a Theorem of Gauss-Kuzmin-Lévy.'' *Arch. Math.* **58**, 251-256, 1992.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/kuzmin/kuzmin.html

Flajolet, P. and Vallée, B. ``On the Gauss-Kuzmin-Wirsing Constant.'' Unpublished memo. 1995. http://pauillac.inria.fr/algo/flajolet/Publications/gauss-kuzmin.ps.

Knuth, D. E. *The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed.*
Reading, MA: Addison-Wesley, 1981.

MacLeod, A. J. ``High-Accuracy Numerical Values of the Gauss-Kuzmin Continued Fraction Problem.'' *Computers
Math. Appl.* **26**, 37-44, 1993.

Wirsing, E. ``On the Theorem of Gauss-Kuzmin-Lévy and a Frobenius-Type Theorem for Function Spaces.'' *Acta
Arith.* **24**, 507-528, 1974.

© 1996-9

1999-05-25