## Machin-Like Formulas

Machin-like formulas have the form

 (1)

where , , and are Positive Integers and and are Nonnegative Integers. Some such Formulas can be found by converting the Inverse Tangent decompositions for which in the table of Todd (1949) to Inverse Cotangents. However, this gives only Machin-like formulas in which the smallest term is .

Maclaurin-like formulas can be derived by writing

 (2)

and looking for and such that
 (3)

so
 (4)

Machin-like formulas exist Iff (4) has a solution in Integers. This is equivalent to finding Integer values such that
 (5)

is Real (Borwein and Borwein 1987, p. 345). An equivalent formulation is to find all integral solutions to one of
 (6)

 (7)

for , 5, ....

There are only four such Formulas,

 (8) (9) (10) (11)

known as Machin's Formula, Euler's Machin-Like Formula, Hermann's Formula, and Hutton's Formula. These follow from the identities
 (12) (13) (14) (15)

Machin-like formulas with two terms can also be generated which do not have integral arc cotangent arguments such as Euler's

 (16)

(Wetherfield 1996), and which involve inverse Square Roots, such as
 (17)

Three-term Machin-like formulas include Gauss's Machin-Like Formula

 (18)

Strassnitzky's Formula
 (19)

and the following,
 (20) (21) (22) (23) (24)

The first is due to Størmer, the second due to Rutherford, and the third due to Dase.

Using trigonometric identities such as

 (25)

it is possible to generate an infinite sequence of Machin-like formulas. Systematic searches therefore most often concentrate on formulas with particularly nice'' properties (such as efficiency'').

The efficiency of a Formula is the time it takes to calculate with the Power series for arctangent

 (26)

and can be roughly characterized using Lehmer's measure'' formula
 (27)

The number of terms required to achieve a given precision is roughly proportional to , so lower -values correspond to better sums. The best currently known efficiency is 1.51244, which is achieved by the 6-term series

 (28)
discovered by C.-L. Hwang (1997). Hwang (1997) also discovered the remarkable identities
 (29)
where , , , , and are Positive Integers, and

 (30)

The following table gives the number of Machin-like formulas of terms in the compilation by Wetherfield and Hwang. Except for previously known identities (which are included), the criteria for inclusion are the following:

1. first term digits: measure .

2. first term = 8 digits: measure .

3. first term = 9 digits: measure .

4. first term =10 digits: measure .

 1 1 0 2 4 1.85113 3 106 1.78661 4 39 1.58604 5 90 1.63485 6 120 1.51244 7 113 1.54408 8 18 1.65089 9 4 1.72801 10 78 1.63086 11 34 1.6305 12 188 1.67458 13 37 1.71934 14 5 1.75161 15 24 1.77957 16 51 1.81522 17 5 1.90938 18 570 1.87698 19 1 1.94899 20 11 1.95716 21 1 1.98938 Total 1500 1.51244

See also Euler's Machin-Like Formula, Gauss's Machin-Like Formula, Gregory Number, Hermann's Formula, Hutton's Formula, Inverse Cotangent, Machin's Formula, Pi, Størmer Number, Strassnitzky's Formula

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 347-359, 1987.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Castellanos, D. The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 241-248, 1996.

Hwang, C.-L. More Machin-Type Identities.'' Math. Gaz., 120-121, March 1997.

Lehmer, D. H. On Arccotangent Relations for .'' Amer. Math. Monthly 45, 657-664, 1938.

Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.

Lewin, L. Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.

Nielsen, N. Der Euler'sche Dilogarithms. Leipzig, Germany: Halle, 1909.

Størmer, C. Sur l'Application de la Théorie des Nombres Entiers Complexes à la Solution en Nombres Rationels , , ..., , , ..., de l'Equation....'' Archiv for Mathematik og Naturvidenskab B 19, 75-85, 1896.

Todd, J. A Problem on Arc Tangent Relations.'' Amer. Math. Monthly 56, 517-528, 1949.

Weisstein, E. W. Machin-Like Formulas.'' Mathematica notebook MachinFormulas.m.

Wetherfield, M. The Enhancement of Machin's Formula by Todd's Process.'' Math. Gaz. 80, 333-344, 1996.

Wetherfield, M. Machin Revisited.'' Math. Gaz., 121-123, March 1997.