Related papers: A BBP-style computation for $\pi$ in base 5
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
In this paper we consider integers in base 10 like $abc$, where $a$, $b$, $c$ are digits of the integer, such that $abc^2 - (abc \cdot cba) \; = \; \pm n^2$, where $n$ is a positive integer, as well as equations $abc^2 - (abc \cdot cba) \;…
A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite…
This paper presents a decision-theoretic approach to statistical inference that satisfies the likelihood principle (LP) without using prior information. Unlike the Bayesian approach, which also satisfies LP, we do not assume knowledge of…
In terms of the hypergeometric method, we give the extensions of two known series for $\pi$. Further, other twenty-nine summation formulas for $\pi$, $\pi^2$ and $1/\pi$ with free parameters are also derived in the same way.
A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible…
Let $k$ be a finite field extension of the function field $\bfF_p(T)$ and $\bar{k}$ its algebraic closure. We count points in projective space $\Bbb P ^{n-1}(\bar{k})$ with given height and of fixed degree $d$ over the field $k$. If…
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…
In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas-Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the…
The unit cost model is both convenient and largely realistic for describing integer decision algorithms over (+,*). Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware,…
In this paper we give an algorithm for computing the mth base-b digit (m=1 is the least significant digit) of an integer n (actually, it finds sharp approximations to n/b^m mod 1), where n is defined as the last number in a sequence of…
We will generalize the combinatorial algorithms for computing $\pi(x)$ to compute sums ${F(x) = \sum_{p \leq x} p^k}$ for $k \in \mathbb{Z}_{\geq 0}$. The detailed exposition of algorithms is included along with implementation details.
Completing Loo-Keng Hua's approach to the real number system pioneered in 1962, this paper defines arithmetical operations directly on infinite decimals without appealing to any ordering structure. Therefore, the widespread belief that…
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…
We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \pi, in Logspace. Our main result is that…
There is little known about the methods used by the ancient Babylonians and Egyptians to arrive at their recorded estimates of the value of Pi. A surprisingly accurate estimate of Pi was recently revealed coded within a verse in the book of…
Given an integer $b\geqslant 2$ and a set $P$ of prime numbers, the set $T_P $ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n)_{n\geqslant 1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and…
A real number $x$ is absolutely normal if, for every base $b\ge 2$, every two equally long strings of digits appear with equal asymptotic frequency in the base-$b$ expansion of $x$. This paper presents an explicit algorithm that generates…
We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The…
Interpreting samples from likelihood or posterior probability density functions is rarely as straightforward as it seems it should be. Producing publication-quality graphics of these distributions is often similarly painful. In this short…