Related papers: A new method for computing number PI
An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals.…
We obtained a new formula for $\pi$.
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 a previous paper a new approach has been introduced for computing, recursively and numerically, one-loop tensor integrals. Here we describe a few modifications of the original method that allow a more efficient numerical implementation…
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…
In this paper we prove some new series for $1/\pi$ as well as related congruences. We also raise several new kinds of series for $1/\pi$ and present some related conjectural congruences involving representations of primes by binary…
A re-calculation of a known family of formulas of PI is carried out, revisiting the old Archimedes' algorithm. This allows to identify a general family equation and three new simple formulas of Pi in terms of the golden ratio PHI in the…
In this paper, we outline an approach to verifying parallel programs. A new mathematical model of parallel programs is introduced. The introduced model is illustrated by the verification of the matrix multiplication MPI program.
A method for computing probabilistic propositions is presented. It assumes the availability of a single external routine for computing the probability of one instantiated variable, given a conjunction of other instantiated variables. In…
In this paper we establish a number of new estimates concerning the prime counting function \pi(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in…
An efficient numerical algorithm for the computation of linking number is presented. The algorithm keep tracks or rounding error so that it can ensure the correctness of the results.
We have rediscovered a simple algorithm to compute the mathematical constant \[ \pi=3.14159265\cdots. \] The algorithm had been known for a long time but it might not be recognized as a fast, practical algorithm. The time complexity of it…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
There are several reformulations of the Vi\`ete's formula for pi that have been reported in the modern literature. In this paper we show another analog to the Vi\`ete's formula for pi by Chebyshev polynomials of the first kind.
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…
We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit $1/\pi$. We will not use the modular theory nor either the Gauss'…
We describe a new algorithm that computes the n-th Bernoulli number in n^(4/3 + o(1)) bit operations. This improves on previous algorithms that had complexity n^(2 + o(1)).
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 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…