Jonathan Sorenson
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Analysis of Algorithms for Moser's Problems on Sums of Consecutive Primes Open
In his 1963 paper on the sum of consecutive primes, Moser posed four open questions related to $f(n)$, the number of ways an integer $n$ can be written as a sum of consecutive primes. (See also problem C2 from Richard K.~Guy's \textit{Unso…
Explicit Bounds and Parallel Algorithms for Counting Multiply Gleeful Numbers Open
Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful…
An algorithm and computation to verify Legendre’s conjecture up to $$7\cdot 10^{13}$$ Open
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…
Reducing the Space Used by the Sieve of Eratosthenes When Factoring Open
We present a version of the sieve of Eratosthenes that can factor all integers $\le x$ in $O(x \log\log x)$ arithmetic operations using at most $O(\sqrt{x}/\log\log x)$ bits of space. This is an improved space bound under the condition tha…
An algorithm and computation to verify Legendre's Conjecture up to $3.33\cdot10^{13}$ Open
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 sub…
An Algorithm for Ennola's Second Theorem and Counting Smooth Numbers in Practice Open
Let $Ψ(x,y)$ count the number of positive integers $n\le x$ such that every prime divisor of $n$ is at most $y$. Given inputs $x$ and $y$, what is the best way to estimate $Ψ(x,y)$? We address this problem in three ways: with a new algorit…
Computation of the least primitive root Open
Let $g(p)$ denote the least primitive root modulo $p$, and $h(p)$ the least primitive root modulo $p^2$. We computed $g(p)$ and $h(p)$ for all primes $p\le 10^{16}$. Here we present the results of that computation and prove three theorems …
An Algorithm to Find Sums of Powers of Consecutive Primes Open
We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant ti…
AtomNet PoseRanker: Enriching Ligand Pose Quality for Dynamic Proteins in Virtual High Throughput Screens Open
Structure-based, virtual High Throughput Screening (vHTS) methods for predicting ligand activity in drug discovery are important when there are no or relatively few known compounds that interact with a therapeutic target of interest. State…
Algorithms to Uniformly Generate Random Factored Smooth Integers Open
Let $x\ge y>0$ be integers. A positive integer is $y$-smooth if all its prime divisors are at most $y$. Let $Ψ(x,y)$ count the number of $y$-smooth integers up to $x$. We present several algorithms that will generate an integer $n\le x$ at…
Two algorithms to find primes in patterns Open
Let $k\\ge 1$ be an integer, and let $P= (f_1(x), \\ldots, f_k(x) )$ be $k$\nadmissible linear polynomials over the integers, or \\textit{the pattern}. We\npresent two algorithms that find all integers $x$ where $\\max{ \\{f_i(x) \\} }\n\\…
Preface Open
The biennial, international Algorithmic Number Theory Symposium (ANTS) provides the premier international forum for state-of-the-art research in computational and algorithmic number theory.This conference is devoted to algorithmic aspects …
Strong pseudoprimes to twelve prime bases Open
Let $\\psi_m$ be the smallest strong pseudoprime to the first $m$ prime bases.\nThis value is known for $1 \\leq m \\leq 11$. We extend this by finding\n$\\psi_{12}$ and $\\psi_{13}$. We also present an algorithm to find all integers\n$n\\…