相关论文: A new form of the Machin-like formula for pi by it…
We present a simple recurrent formula to generate the Machin-like expression for calculating $\pi/4$. The method works for any denominator in the starting term and always provides a finite decomposition. We show that the terms in the…
In this paper we present a two-term Machin-like formula for pi \[\frac{\pi}{4} = 2^{k - 1}\arctan\left(\frac{1}{u_1}\right) + \arctan\left(\frac{1}{u_2}\right)\] with small Lehmer's measure $e \approx 0.245319$ and describe iteration…
Previously we have proposed a new method of transforming quotients into integer reciprocals in the Machin-like formulas for $\pi$. As a further development, here we show how to generate a multi-term Machin-like formula for $\pi$ with a…
Lehmer defined a measure $$ \mu=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|\beta_j\right|\right)}, $$ where the $\beta_j$ may be either integers or rational numbers in a Machin-like formula for $\pi$. When the $\beta_j$ are integers,…
In this work, we develop a new iterative method for computing the digits of $\pi$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $\pi$ with squared convergence that we…
In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $\pi$ by using its rational approximation. In this approximation, both terms are constructed by using a…
In this paper we propose a new method for determination of the two-term Machin-like formula for pi with arbitrarily small arguments of the arctangent function. This approach excludes irrational numbers in computation and leads to a…
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…
In our recent publication we have proposed a new methodology for determination of the two-term Machin-like formula for pi with small arguments of the arctangent function of kind $$ \frac{\pi }{4} = {2^{k - 1}}\arctan \left(…
In this work, we obtain an iterative formula that can be used for computing digits of $\pi$ and nested radicals of kind $c_n/\sqrt{2 - c_{n - 1}}$, where $c_0 = 0$ and $c_n = \sqrt{2 + c_{n - 1}}$. We also show how with the help of this…
We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated $2$-integers, that is values of the form $b/2^a$ for some integers $a$ and~$b$.…
Machin-like arctangent relations are classical tools for computing $\pi$, with efficiency quantified by the Lehmer measure ($\lambda$). We present a framework for discovering low-measure relations by coupling the PSLQ integer-relation…
The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…
We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \frac{\pi}{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction…
The prime-counting function $\pi(x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $\pi(x)$ with time…
In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as…
An application of (iterated) Bauer-Muir acceleration can give an Ap\'ery-like continued fraction for $\pi$ with irrational coefficients, and much faster convergence. It can be considered a generalized continued fraction with the same matrix…
A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.
We present an improved version of the analytic method for calculating $\pi(x)$, the number of prime numbers not exceeding $x$. We implemented this method in cooperation with J. Franke, T. Kleinjung and A. Jost and calculated the value…
This paper describes recent advances in the combinatorial method for computing $\pi(x)$, the number of primes $\leq x$. In particular, the memory usage has been reduced by a factor of $\log x$, and modifications for shared- and…