Pierce Expansion
The Pierce expansion, or alternated Egyptian product, of a real number 0<x<1 is the unique increasing sequence {a_1,a_2,...} of positive integers a_i such that
A number 0<x<1 has a finite Pierce expansion iff x is rational.
Special cases are summarized in the following table.
If x is of the form
| [画像: x=(c-sqrt(c^2-4))/2, ] |
(2)
|
then there is a closed-form for the Pierce expansion given by
| x={c_0-1,c_0+1,c_1-1,c_1+1,c_2-1,c_2+1,...}, |
(3)
|
where
and c_(k+1)=c_k^3-3c_k (Shallit 1984). This recurrence has explicit solution
| c_k^((c))=-2cos[3^kcos^(-1)(-1/2c)] |
(6)
|
not noted by Shallit (1984).
c=3, corresponding to x=(3-sqrt(5))/2, has the particularly beautiful form
where F_n is a Fibonacci number.
The following table gives coefficients c_k and a_k for some small integer c.
See also
Engel ExpansionExplore with Wolfram|Alpha
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References
Erdős, P. and Shallit, J. O. "New Bounds on the Length of Finite Pierce and Engel Series." Sem. Theor. Nombres Bordeaux 3, 43-53, 1991.Keselj, V. "Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations." Sep. 10, 1996. http://www.cs.uwaterloo.ca/research/tr/1996/21/cs-96-21.pdf.Mays, M. E. "Iterating the Division Algorithm." Fib. Quart. 25, 204-213, 1987.Pierce, T. A. "On an Algorithm and Its Use in Approximating Roots of Polynomials." Amer. Math. Monthly 36, 523-525, 1929.Salzer, H. E. "The Approximation of Numbers as Sums of Reciprocals." Amer. Math. Monthly 54, 135-142, 1947.Shallit, J. O. "Some Predictable Pierce Expansions." Fib. Quart. 22, 332-335, 1984.Shallit, J. O. "Metric Theory of Pierce Expansions." Fib. Quart. 24, 22-40, 1986.Sloane, N. J. A. Sequences A001999/M3055, A006275/M1342, A006283/M3092, A006284/M1593, A006276/M1298, A020725, A091831, A091846, A112845, A118242, and A132201 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Pierce ExpansionCite this as:
Weisstein, Eric W. "Pierce Expansion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PierceExpansion.html