arXiv subjects

Jonathan Webster

Publications and source records attributed to Jonathan Webster.

13 recordsLinked to original sources

arXiv2025

Algorithms for Carmichael numbers

Our primary concern is the computational complexity of algorithms that find all Carmichael numbers less than some specified bound $B$. We have three related results. First, we show CARMICHAELS is in $\textbf{P}$, where only the run-time is conditioned on the ERH. Second, we state a heuristically optimal tabulation algorithm, which is the first asymptotic improvement to tabulation algorithms in the $50$ years since Swift first described the prime-by-prime approach. Third, we implemented a related algorithm that tabulated $100$ times further while only doing about $5$ times the work of the prior tabulation. We found $308,279,939$ Carmichael numbers less than $10^{24}$ and we provide some statistics on these numbers.

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arXiv2024

Advances in Tabulating Carmichael Numbers

We report that there are $49679870$ Carmichael numbers less than $10^{22}$ which is an order of magnitude improvement on Richard Pinch's prior work. We find Carmichael numbers of the form $n = Pqr$ using an algorithm bifurcated by the size of $P$ with respect to the tabulation bound $B$. For $P < 7 \cdot 10^7$, we found $35985331$ Carmichael numbers and $1202914$ of them were less than $10^{22}$. When $P > 7 \cdot 10^7$, we found $48476956$ Carmichael numbers less than $10^{22}$. We provide a comprehensive overview of both cases of the algorithm. For the large case, we show and implement asymptotically faster ways to tabulate compared to the prior tabulation. We also provide an asymptotic estimate of the cost of this algorithm. It is interesting that Carmichael numbers are worst case inputs to this algorithm. So, providing a more robust asymptotic analysis of the cost of the algorithm would likely require resolution of long-standing open questions regarding the asymptotic density of Carmichael numbers.

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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.

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arXiv2023

Tabulating Absolute Lucas Pseudoprimes

In 1977, Hugh Williams studied Lucas pseudoprimes to all Lucas sequences of a fixed discriminant. These are composite numbers analogous to Carmichael numbers and they satisfy a Korselt-like criterion: $n$ must be a product of distinct primes and $p_i - \delta_{p_i} | n - \delta_n $ where $\delta_n$ is a Legendre symbol with the first argument being the discriminant of the Lucas sequence. Motivated by tabulation algorithms for Carmichael numbers, we give algorithms to tabulate these numbers and provide some asymptotic analysis of the algorithms. We show that there are only finitely many absolute Lucas pseudoprimes $n = \prod_{i = 1}^k p_i$ with a given set of $k-2$ prime factors. We also provide the first known tabulation for discriminant $5$.

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arXiv2021

Kurt Hensel on Common Inessential Discriminant Divisors, 1894

The problem of the "common inessential discriminant divisors" attracted the attention of Dedekind, Kronecker, and Hensel in the early days of algebraic number theory. Four sources are particularly important: Dedekind's announcement, in 1871, of the second edition of Dirichlet's lectures \cite{anzeige}, Dedekind's 1878 paper, the 25th section of Kronecker's 1882 \textit{Grundz\"uge}, and Hensel's 1894 paper, which is our focus here. Both of the key papers of Dedekind were translated and annotated in our paper "Dedekind on higher congruences and index divisors, 1871 and 1878." (arXiv:2107.08905}. We here present an annotated translation of Kurt Hensel's "Arithmetische Untersuchungen \"uber die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung" (\textit{Journal f\"ur die Reine und Angewandte Mathematik}, \textbf{113} (1894), 128--160).

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arXiv2021

Dedekind on Higher Congruences and Index Divisors, 1871 and 1878

Dedekind's theorem connecting ideal theory and polynomial congruences appears in all textbooks on algebraic number theory, but few books note its connection to the problem of ``common index divisors.'' As part of a project to study the history of this problem, we present an annotated translation of two of Dedekind's papers on the subject: a notice about the first publication of Dedekind's ideal theory in 1871 and a paper of 1878 giving proofs of the results announced in 1871 and giving a necessary and sufficient condition for the existence of common index divisors. A separate paper will analyze Hensel's 1894 paper containing the same theorem.

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arXiv2019

Algorithms for the Multiplication Table Problem

Let $M(n)$ denote the number of distinct entries in the $n \times n$ multiplication table. The function $M(n)$ has been studied by Erd\H{o}s, Tenenbaum, Ford, and others, but the asymptotic behaviour of $M(n)$ as $n \to \infty$ is not known precisely. Thus, there is some interest in algorithms for computing $M(n)$ either exactly or approximately. We compare several algorithms for computing $M(n)$ exactly, and give a new algorithm that has a subquadratic running time. We also present two Monte Carlo algorithms for approximate computation of $M(n)$. We give the results of exact computations for values of $n$ up to $2^{30}$, and of Monte Carlo computations for $n$ up to $2^{100,000,000}$, and compare our experimental results with Ford's order-of-magnitude result.

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arXiv2019

An Algorithm and Estimates for the Erd\H{o}s-Selfridge Function (work in progress)

Let $p(n)$ denote the smallest prime divisor of the integer $n$. Define the function $g(k)$ to be the smallest integer $>k+1$ such that $p(\binom{g(k)}{k})>k$. So we have $g(2)=6$ and $g(3)=g(4)=7$. In this paper we present the following new results on the Erd\H{o}s-Selfridge function $g(k)$: We present a new algorithm to compute the value of $g(k)$, and use it to both verify previous work and compute new values of $g(k)$, with our current limit being $$ g(323)= 1\ 69829\ 77104\ 46041\ 21145\ 63251\ 22499. $$ We define a new function $\hat{g}(k)$, and under the assumption of our Uniform Distribution Heuristic we show that $$ \log g(k) = \log \hat{g}(k) + O(\log k) $$ with high "probability". We also provide computational evidence to support our claim that $\hat{g}(k)$ estimates $g(k)$ reasonably well in practice. There are several open conjectures on the behavior of $g(k)$ which we are able to prove for $\hat{g}(k)$, namely that $$ 0.525\ldots +o(1) \quad \le \quad \frac{\log \hat{g}(k)}{k/\log k} \quad \le \quad 1+o(1), $$ and that $$ \limsup_{k\rightarrow\infty} \frac{\hat{g}(k+1)}{\hat{g}(k)}=\infty.$$ Let $G(x,k)$ count the number of integers $n\le x$ such that $p(\binom{n}{k})>k$. Unconditionally, we prove that for large $x$, $G(x,k)$ is asymptotic to $x/\hat{g}(k)$. And finally, we show that the running time of our new algorithm is at most $g(k) \exp[ -c (k\log\log k) /(\log k)^2 (1+o(1))]$ for a constant $c>0$.

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arXiv2018

Two Algorithms to Find Primes in Patterns

Let $k\ge 1$ be an integer, and let $P= (f_1(x), \ldots, f_k(x) )$ be $k$ admissible linear polynomials over the integers, or \textit{the pattern}. We present two algorithms that find all integers $x$ where $\max{ \{f_i(x) \} } \le n$ and all the $f_i(x)$ are prime. Our first algorithm takes at most $O_P(n/(\log\log n)^k)$ arithmetic operations using $O(k\sqrt{n})$ space. Our second algorithm takes slightly more time, $O_P(n/(\log \log n)^{k-1})$ arithmetic operations, but uses only $n^{1/c}$ space for a constant $c>2$. We prove correctness unconditionally, but the running time relies on two unproven but reasonable conjectures. We are unaware of any previous complexity results for this problem beyond the use of a prime sieve. We also implemented several parallel versions of our second algorithm to show it is viable in practice. In particular, we found some new Cunningham chains of length 15, and we found all quadruplet primes up to $10^{17}$.

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arXiv2018

Fast tabulation of challenge pseudoprimes

We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW challenge pseudoprimes with two or three prime factors up to $2^{80}$. In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods.

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arXiv2015

Strong Pseudoprimes to Twelve Prime Bases

Let $\psi_m$ be the smallest strong pseudoprime to the first $m$ prime bases. This value is known for $1 \leq m \leq 11$. We extend this by finding $\psi_{12}$ and $\psi_{13}$. We also present an algorithm to find all integers $n\le B$ that are strong pseudoprimes to the first $m$ prime bases; with a reasonable heuristic assumption we can show that it takes at most $B^{2/3+o(1)}$ time.

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arXiv2012

Simple cubic function fields and class number computations

In this paper, we study simple cubic fields in the function field setting, and also generalize the notion of a set of exceptional units to cubic function fields, namely the notion of $k$-exceptional units. We give a simple proof that the Galois simple cubic function fields are the immediate analog of Shanks simplest cubic number fields. In addition to computing the invariants, including a formula for the regulator, we compute the class numbers of the Galois simple cubic function fields over $\mathbb{F}_{5}$ and $\mathbb{F}_{7}$ using truncated Euler products. Finally, as an additional application, we determine all Galois simple cubic function fields with class number one, subject to a mild restriction.

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arXiv2010

Computations in Cubic Function Fields of Characteristic Three

This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of these function fields, we are able to calculate the field discriminant and the genus. We also describe the splitting behavior of any place, and give composition and reduction algorithms for arithmetic in the ideal class group.

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