Related papers: The combinatorial algorithm for computing $\pi(x)$
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…
We derive several numerical methods for designing optimized first-order algorithms in unconstrained convex optimization settings. Our methods are based on the Performance Estimation Problem (PEP) framework, which casts the worst-case…
A GPU implementation of an algorithm to compute the Mertens function in O(x2/3+{\ko}) time is discussed. Results for x up to $10^{22}$, and a new extreme value for $M(x)/x^{1/2}$, -0.585768 ($M(x) \approx -1.996 \ast 10^9$ at $x \approx…
After Strassen presented the first sub-cubic matrix multiplication algorithm, many Strassen-like algorithms are presented. Most of them with low asymptotic cost have large hidden leading coefficient which are thus impractical. To reduce the…
Many algorithms have been developed for enumerating various combinatorial objects in time exponentially less than the number of objects. Two common classes of algorithms are dynamic programming and the transfer matrix method. This paper…
An alphabetic binary tree formulation applies to problems in which an outcome needs to be determined via alphabetically ordered search prior to the termination of some window of opportunity. Rather than finding a decision tree minimizing…
This paper presents a novel approach at the intersection of machine learning and number theory, focusing on the classification of prime and non-prime numbers. At the core of our research is the development of a highly sparse encoding…
We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $\pi(x)$.
The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used…
Decimal multiplication is the task of multiplying two numbers in base $10^N.$ Specifically, we focus on the number-theoretic transform (NTT) family of algorithms. Using only portable techniques, we achieve a 3x-5x speedup over the mpdecimal…
We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small…
We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we…
We present PDLP, a practical first-order method for linear programming (LP) designed to solve large-scale LP problems. PDLP is based on the primal-dual hybrid gradient (PDHG) method applied to the minimax formulation of LP. PDLP…
All the known approximations of the number of primes pi(n) not exceeding any given integer n are derived from real-valued functions that are asymptotic to pi(x), such as x/log x, Li(x) and Riemann's function R(x). The degree of…
We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order…
In this paper, we extend the work of (Abbondati et al., 2024) on decoding simultaneous rational number codes by addressing two important scenarios: multiplicities and the presence of bad primes (divisors of denominators). First, we…
A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function $\pi_2(x)$, the number of semiprimes less than or equal to $x$, is studied. Using a combinatorial argument,…
We prove that the complexity of computing the table of primes between $1$ and $n$ on a multitape Turing machine is $O(n \log n)$.
We present a natural, combinatorial problem whose solution is given by the meta-Fibonacci recurrence relation $a(n) = \sum_{i=1}^p a(n-i+1 - a(n-i))$, where $p$ is prime. This combinatorial problem is less general than those given in [3]…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…