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

Author

Listed:
  • Sanjar M. Abrarov

    (Thoth Technology Inc., Algonquin Radio Observatory, Achray Road, RR6, Pembroke, ON K8A 6W7, Canada
    Department Earth and Space Science and Engineering, York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada
    Epic College of Technology, 5670 McAdam Rd., Mississauga, ON L4Z 1T2, Canada)

  • Rajinder K. Jagpal

    (Epic College of Technology, 5670 McAdam Rd., Mississauga, ON L4Z 1T2, Canada
    Department Physics and Astronomy, York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada)

  • Rehan Siddiqui

    (Department Earth and Space Science and Engineering, York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada
    Epic College of Technology, 5670 McAdam Rd., Mississauga, ON L4Z 1T2, Canada
    Department Physics and Astronomy, 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
    Department Earth and Space Science and Engineering, York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada
    Department Physics and Astronomy, York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada)

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 π of the kind π 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.

Suggested Citation

  • Sanjar M. Abrarov & Rajinder K. Jagpal & Rehan Siddiqui & Brendan M. Quine, 2021. "Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for π," Mathematics, MDPI, vol. 9(17), pages 1-20, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2162-:d:629054
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