arXiv subjects

Jonathan Sorenson

Publications and source records attributed to Jonathan Sorenson.

4 recordsLinked to original sources

arXiv2024

An algorithm and computation to verify Legendre's Conjecture up to $3.33\cdot10^{13}$

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre's conjecture claims that for every positive integer $n$, there exists a prime between $n^2$ and $(n+1)^2$. Oppermann's conjecture subsumes Legendre's conjecture by claiming there are primes between $n^2$ and $n(n+1)$ and also between $n(n+1)$ and $(n+1)^2$. Using Cram\'er's conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann's conjecture, and hence also Legendre's conjecture, for all $n\le N$ in time $O( N \log N \log \log N)$ and space $N^{O(1/\log \log N)}$. We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann's conjecture from the previous $N = 2\cdot 10^{9}$ up to $N = 3.33\cdot 10^{13}$, so we were finding $27$ digit primes. The computation ran for about half a year on four Intel Xeon Phi $7210$ processors using a total of $256$ cores.

math.NT↗

arXiv2020

Algorithms to Uniformly Generate Random Factored Smooth Integers

Let $x\ge y>0$ be integers. A positive integer is $y$-smooth if all its prime divisors are at most $y$. Let $\Psi(x,y)$ count the number of $y$-smooth integers up to $x$. We present several algorithms that will generate an integer $n\le x$ at random, with known prime factorization, such that $n$ is $y$-smooth. We begin by describing algorithms to compute $\Psi(x,y)$ exactly and to enumerate $y$-smooth integers up to $x$ in lexicographic order by prime divisor. Both of these are based on Buchstab's identity, and were likely known before. Then we present an algorithm that accepts as input a parameter $r$, $0\le r<1$, and returns the integer $n$ that is at position $\lfloor r\Psi(x,y)\rfloor$ in the lexicographic ordering of all $y$-smooth integers up to $x$. Here position 0 is the first position. Thus, $n$ is generated uniformly so long as $r$ is chosen uniformly. This algorithm has a running time of $O(\Psi(x,y)\log\log y)$ arithmetic operations. We then explore the tradeoff between speed and rigor. By relaxing the uniformity of the output and allowing for multiple heuristics in our runtime analysis, we improve the running time to $$ O\left( \frac{ (\log x)^3 }{\log\log x} \right)$$ arithmetic operations. We conclude with a sample run by generating a $10000$-smooth integer $\le 10^{100}$.

math.NT↗

arXiv2013

Near-Optimal Online Multiselection in Internal and External Memory

We introduce an online version of the multiselection problem, in which q selection queries are requested on an unsorted array of n elements. We provide the first online algorithm that is 1-competitive with Kaligosi et al. [ICALP 2005] in terms of comparison complexity. Our algorithm also supports online search queries efficiently. We then extend our algorithm to the dynamic setting, while retaining online functionality, by supporting arbitrary insertions and deletions on the array. Assuming that the insertion of an element is immediately preceded by a search for that element, we show that our dynamic online algorithm performs an optimal number of comparisons, up to lower order terms and an additive O(n) term. For the external memory model, we describe the first online multiselection algorithm that is O(1)-competitive. This result improves upon the work of Sibeyn [Journal of Algorithms 2006] when q > m, where m is the number of blocks that can be stored in main memory. We also extend it to support searches, insertions, and deletions of elements efficiently.

cs.DS↗

arXiv2013

Approximately counting semismooth integers

An integer $n$ is $(y,z)$-semismooth if $n=pm$ where $m$ is an integer with all prime divisors $\le y$ and $p$ is 1 or a prime $\le z$. arge quantities of semismooth integers are utilized in modern integer factoring algorithms, such as the number field sieve, that incorporate the so-called large prime variant. Thus, it is useful for factoring practitioners to be able to estimate the value of $Ψ(x,y,z)$, the number of $(y,z)$-semismooth integers up to $x$, so that they can better set algorithm parameters and minimize running times, which could be weeks or months on a cluster supercomputer. In this paper, we explore several algorithms to approximate $Ψ(x,y,z)$ using a generalization of Buchstab's identity with numeric integration.

cs.DS↗