Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for ?

Sanjar M. Abrarov, Rajinder K. Jagpal, Rehan Siddiqui and Brendan M. Quine
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Sanjar M. Abrarov: Thoth Technology Inc., Algonquin Radio Observatory, Achray Road, RR6, Pembroke, ON K8A 6W7, Canada
Rajinder K. Jagpal: Epic College of Technology, 5670 McAdam Rd., Mississauga, ON L4Z 1T2, Canada
Rehan Siddiqui: Department Earth and Space Science and Engineering, York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada
Brendan M. Quine: Thoth Technology Inc., Algonquin Radio Observatory, Achray Road, RR6, Pembroke, ON K8A 6W7, Canada

Mathematics, 2021, vol. 9, issue 17, 1-20

Abstract: In our earlier publication we have shown how to compute by iteration a rational number u 2 , k in the two-term Machin-like formula for ? of the kind ? 4 = 2 k ? 1 arctan 1 u 1 , k + arctan 1 u 2 , k , k ? Z , k ? 1 , where u 1 , k can be chosen as an integer u 1 , k = a k / 2 ? a k ? 1 with nested radicals defined as a k = 2 + a k ? 1 and a 0 = 0 . In this work, we report an alternative method for determination of the integer u 1 , k . This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set a k in computation of the integer u 1 , k . Mathematica programs validating these results are presented.

Keywords: constant ?; Machin-like formula; Lehmer’s measure; surd number; Ramanujan’s nested radical (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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