Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for π
| Autor: | Rehan Siddiqui, Rajinder K. Jagpal, Brendan M. Quine, Sanjar M. Abrarov |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Discrete mathematics
Rational number Machin-like formula Ramanujan’s nested radical General Mathematics Computation Term (logic) Lehmer’s measure Set (abstract data type) 11Y60 General Mathematics (math.GM) Simple (abstract algebra) constant π Irrational number FOS: Mathematics QA1-939 Computer Science (miscellaneous) Mathematics - General Mathematics Engineering (miscellaneous) Mathematics surd number Integer (computer science) |
| Zdroj: | Mathematics Volume 9 Issue 17 Mathematics, Vol 9, Iss 2162, p 2162 (2021) |
| ISSN: | 2227-7390 |
| DOI: | 10.3390/math9172162 |
| Popis: | 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 pi of kind $$\frac{\pi}{4}=2^{k-1}\arctan\left(\frac{1}{u_{1,k}}\right)+\arctan\left(\frac{1}{u_{2,k}}\right),\qquad k\in \mathbb{Z},\quad k\ge 1,$$ where ${u_{1,k}}$ can be chosen as an integer ${u_{1,k}} = \left\lfloor{{a_k}/\sqrt{2-a_{k-1}}}\right\rfloor$ with nested radicals defined as ${a_k}=\sqrt{2+a_{k-1}}$ 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 $\left\{a_k\right\}$ in computation of the integer $u_{1,k}$. Mathematica programs validating these results are presented. Comment: 30 pages |
| Databáze: | OpenAIRE |
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