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Fred B. Holt

Publications and source records attributed to Fred B. Holt.

At least 19 recordsLinked to original sources
arXiv2026

Discrete dynamics of Eratosthenes sieve

We study Eratosthenes sieve as a discrete dynamic system. At each stage of the sieve there is a cycle of gaps ${\mathcal G}(p^\#)$ of length $ϕ(p^\#)$ and span $p^\#$. There is a recursion ${\mathcal G}(p_k^\#)\longrightarrow {\mathcal G}(p_{k+1}^\#)$ that creates the next cycle from the current one. If we take initial conditions from the cycle ${\mathcal G}(p_0^\#)$, then for all constellations of span $|s| < 2p_1$, including gaps $g < 2p_1$, the driving terms of various lengths form Markov chains. These yield {\it exact} models for the populations $n_s(p_k^\#)$ for all further stages of the sieve. If $s$ is an admissible constellation of length $J$, then its population $n_{s,J}(p^\#)$ grows as $Θ\left( \prod (q-J-1)\right)$. So we factor out the superexponential growth to obtain the exact model for the relative population $w_s(p_k^\#)$ of the constellation $s$ across all further stages of the sieve. $$ w_{s,J}(p_k^\#) \; = \; n_{s,J}(p_k^\#) \, / \, \prod_{J+1 < p \le p_k} (p-J-1) $$ The asymptotic value of the relative population is a constant ${w_{s,J}(\infty) \ge 1}$ that depends only on the odd prime factors that divide a span in $s$. Assuming that the instances of a constellation $s$ are approximately uniformly distributed in ${\mathcal G}(p_k^\#)$, we develop first-order estimates of the number of instances $s$ that would occur in the interval of survival $ΔH(p_k) = (p_k^2, p_{k+1}^2]$. We define a statistic $η_s(p_k)$, the quadratic density of the constellation $s$ over the interval $ΔH(p_k)$. We show that the first-order estimates $\widehat{η_g}(p)$ for prime gaps agree with samples up to $5.677\,E14$.

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arXiv2026

On nonconvex constellations among primes II: (458,3240)

Extending our work on the $k$-tuple conjecture, we previously applied those methods to the Engelsma counterexamples (narrow constellations) of length $J=459$ and span $|s|=3242$. Here we extend that analysis to the $116$ Engelsma counterexamples of length $J=458$ and $|s|=3240$. We track the evolution of these $116$ counterexamples from inadmissible driving terms starting in the cycle of gaps ${\mathcal G}(11^\#)$ up through their first appearance in ${\mathcal G}(113^\#)$. We continue developing primorial coordinates for each admissible instance through a breadth-first exhaustive search through ${\mathcal G}(211^\#)$. Each of the $(458,3240)$ constellations sits inside a $(459,3242)$ constellation, which we call its {\em parent}. We show that no $(458,3240)$ constellation occurs outside of its parent until the cycle ${\mathcal G}(227^\#)$. The early evolution of the $(458,3240)$ constellations is dominated by the evolution of their parents, which we have previously studied. For each $(458,3240)$-counterexample we calculate its asymptotic relative population, among other constellations of length $J=458$.

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arXiv2026

On nonconvex constellations among primes I

Extending our work on the $k$-tuple conjecture, we apply those methods to the Engelsma counterexamples (narrow constellations) of length $J=459$ and span $|s|=3242$. We track the evolution of these $58$ counterexamples from inadmissible driving terms starting in the cycle of gaps ${\mathcal G}(11^\#)$ up through their first appearance in ${\mathcal G}(113^\#)$. We continue developing primorial coordinates for each admissible instance through a breadth-first exhaustive search through ${\mathcal G}(211^\#)$, at which point we need to develop strategies for depth-first searches for an instance that would survive Eratosthenes sieve. Our calculations show that {\em none} of the $(459,3242)$-counterexamples occur before $9.7\,E73$. For each of the $58$ Engelsma $(459,3242)$-counterexamples we calculate its asymptotic relative population, among other constellations of length $J=459$, and we study how these counterexamples work. In this version (9 April) we have completed the calculations in Table 6 to include all of the terms in primorial expansion for the smallest initial generator and corrected a typographical error on page 6.

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arXiv2026

Surviving Eratosthenes sieve I: quadratic density and Legendre's conjecture

We have been studying Eratosthenes sieve as a discrete dynamic system, obtaining exact models for the relative populations for small gaps (currently gaps $g \le 82$) in the cycle of gaps ${\mathcal G}(p^\#)$ at each stage of the sieve. The gaps in the interval $ΔH(p_k)=[p_k^2, p_{k+1}^2]$ are fixed in ${\mathcal G}(p^\#)$ and survive all subsequent stages of the sieve to be confirmed as gaps between primes. We have shown that samples of gaps between primes over these intervals of survival $ΔH(p_k)$ have population distributions that reflect the relative population models $w_g(p_k^\#)$. This paper advances our study of the estimates of survival across stages of the sieve. Inspired by Legendre's conjecture, we introduce the concept of quadratic density $η_s(p_k)$, which is the expected population of the constellation $s$ in the intervals $[n^2, (n+1)^2]$ for $p_k \le n < p_{k+1}$. We show that once a gap occurs in ${\mathcal G}(p^\#)$, its expected quadratic density increases across all subsequent stages of the sieve. Regarding Legendre's conjecture, beyond postulating one prime in the interval $[n^2,(n+1)^2]$, the quadratic density predicts the populations of several prime gaps within this interval.

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arXiv2025

Eratosthenes sieve supports the $k$-tuple conjecture

Viewing Eratosthenes sieve as a discrete dynamic system, we show that every admissible instance of every admissible constellation of gaps arises and persists in Eratosthenes sieve. For an admissible constellation of length J, we show that its population across stages of the sieve is consistent with the Hardy and Littlewood estimates from 1923. This work strongly connects Eratosthenes sieve to the k-tuple conjecture, and it provides a compact notation, primorial coordinates, for tracking the locations of admissible instances for a constellation.

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arXiv2024

Expected biases in the distribution of consecutive primes

In 2016 Lemke Oliver and Soundararajan examined the gaps between the first hundred million primes and observed biases in their distributions modulo 10. Given our work on the evolution of the populations of various gaps across stages of Eratosthenes sieve, the observed biases are totally expected. The biases observed by Lemke Oliver and Soundararajan are a wonderful example for contrasting the computational range with the asymptotic range for the populations of the gaps between primes. The observed biases are the combination of two phenomena: (a) very small gaps, say $2 \le g \le 30$, get off to quick starts and over the first 100 million primes larger gaps are too early in their evolution; and (b) the assignment of small gaps across the residue classes disadvantages some of those classes - until enormous primes, far beyond the computational range. For modulus 10 and a few other bases, we aggregate the gaps by residue class and track the evolution of these teams as Eratosthenes sieve continues. The relative populations across these teams start with biases across the residue classes. These initial biases fade as the sieve continues. The OS enumeration strongly agrees with a uniform sampling at the corresponding stage of the sieve. The biases persist well beyond the computational range, but they are ultimately transient.

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arXiv2024

On the counts of p-rough numbers

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function $Φ(x,p)$, and we introduce the function $ΔΦ(x,p)$ which is the difference between $Φ(x,p)$ and the line of symmetry. $ΔΦ(x,p)$ is periodic and bounded and has a rotational symmetry.

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arXiv2023

Models for gaps $g=2p_1$

We have shown previously that at each stage of Eratosthenes sieve there is a corresponding cycle of gaps $\mathcal{G}(p_0^\#)$. We can view these cycles of gaps as a discrete dynamic system, and from this system we can obtain exact models for the populations and relative populations of gaps $g < 2p_1$ if we can get the initial conditions from $\mathcal{G}(p_0^\#)$. In this addendum we have shown that we can produce the model for $g=2p_1$ from these initial conditions. This model requires one special iteration to track the count from $\mathcal{G}(p_0^\#)$ to $\mathcal{G}(p_1^\#)$, after which we can use the general model for these populations. As a specific example we exhibit the model for the gap $g=82$ using $\mathcal{G}(37^\#)$ for initial conditions. We show further that in order to produce the models for $g=2p_1+2$ and beyond from initial conditions in $\mathcal{G}(p_0^\#)$, we would have to track subpopulations of the driving terms until the general model applies, that is until $g < 2p_{k+1}$. This work serves as an addendum to the existing references "Patterns among the Primes" and "Combinatorics of the gaps between primes". We do not duplicate that background here, beyond summarizing a few needed results.

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arXiv2016

On the last digits of consecutive primes

Recently Oliver and Soundararajan made conjectures based on computational enumerations about the frequency of occurrence of pairs of last digits for consecutive primes. By studying Eratosthenes sieve, we have identified discrete dynamic systems that exactly model the populations of gaps across stages of Eratosthenes sieve. Our models provide some insight into the observed biases in the occurrences of last digits in consecutive primes, and the models suggest that the biases will ultimately be reversed for large enough primes. The exact model for populations of gaps across stages of Eratosthenes sieve provides a constructive complement to the probabilistic models rooted in the work of Hardy and Littlewood, and it provides time constants that describe the evolution of the populations of larger gaps.

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arXiv2015

Combinatorics of the gaps between primes

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. The populations of gaps and constellations across stages of Eratosthenes sieve are modeled exactly by discrete dynamic systems. These models and their asymptotic behaviors provide evidence on a number of open problems regarding gaps between prime numbers. For Eratosthenes sieve we show that the analogue of Polignac's conjecture is true: every gap $g=2k$ does occur in the sieve, and its asymptotic population supports the estimates made in Hardy and Littlewood's Conjecture B. A stronger form of Polignac's conjecture also holds for the sieve: for any gap $g=2k$, every feasible constellation $g,g,\ldots,g$ occurs; these constellations correspond to consecutive primes in arithmetic progression. The models also provide evidence toward resolving a series of questions posed by Erdös and Turán.

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arXiv2015

Constellations of gaps in Eratosthenes sieve

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. Over the last year we identified a discrete linear system that exactly models the population of any gap across all stages of the sieve. In August 2014 we summarized our results from analyzing this discrete model on populations of single gaps. This paper extends the discrete system to model the populations of constellations of gaps. The most remarkable result is a strong Polignac result on arithmetic progressions. We had previously established that the equivalent of Polignac's conjecture holds for Eratosthenes sieve -- that every even number arises as a gap in the sieve, and its population converges toward the ratio implied by Hardy and Littlewood's Conjecture B. Extending that work to constellations, we here establish that for any even gap $g$, if $p$ is the maximum prime such that $p\# \; | g$ and $P$ is the next prime larger than $p$, then for every $2 \le j_1 < P-1$, the constellation $g,g,\ldots,g$ of length $j_1$ arises in Eratosthenes sieve. This constellation corresponds to an arithmetic progression of $j_1+1$ consecutive candidate primes.

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arXiv2015

Nonsimplicities and the perturbed wedge

In 2010 Santos described the construction of a counterexample to the Hirsch conjecture, and in 2012 Santos and Weibel provided the coordinates for the 40 facets of a 20-dimensional counterexample. In this paper we explore technical details of the construction using Santos and Weibel's work as the motivating example. Santos presented the construction in the dual setting. Here we return to the primal setting, in which Santos' construction calls for repeated application of a perturbed wedge operation, a wedge over a facet followed by a perturbation of one or more other facets. We show that the starting point for the construction is a counterexample "P5" to the nonrevisiting conjecture in dimension 5. However, this polytope P5 is not a simple polytope; it contains two nonsimple vertices. As we repeatedly apply the perturbed wedge, the nonsimplicities grow in dimension while their excess is reduced. Finally in dimension 20, the resulting polytope is simple and its diameter exceeds the Hirsch bound by 1. These notes are a technical companion to the work of Santos and Weibel.

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arXiv2014

Eratosthenes sieve and the gaps between primes

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. By studying this recursion on the cycles of gaps across stages of Eratosthenes sieve, we are able to provide evidence on a number of open problems regarding gaps between prime numbers. The basic counts of short constellations in the cycles of gaps provide evidence toward the twin prime conjecture and toward resolving a series of questions posed by Erdos and Turan. The dynamic system underlying the recursion provides evidence toward Polignac's conjecture and in support of the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.

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arXiv2014

On Polignac's Conjecture

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. As the recursion proceeds, adjacent gaps within longer constellations are added together to produce shorter constellations of the same sum. These additions or closures correspond to removing composite numbers that are divisible by the prime for that stage of Eratosthenes sieve. Although we don't know where in the cycle of gaps a closure will occur, we can enumerate exactly how many copies of various constellations will survive each stage. In this paper, we broaden our study of these systems of constellations of a fixed sum. By generalizing our methods, we are able to demonstrate that for every even number $2n$ the gap $g=2n$ occurs infinitely often through the stages of Eratosthenes sieve. Moreover, we show that asymptotically the ratio of the number of gaps $g=2n$ to the number of gaps $g=2$ at each stage of Eratosthenes sieve converges to the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.

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arXiv2014

On small gaps among primes

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. As the recursion proceeds, adjacent gaps within longer constellations are added together to produce shorter constellations of the same sum. These additions or closures correspond to removing composite numbers that are divisible by the prime for that stage of Eratosthenes sieve. Although we don't know where in the cycle of gaps a closure will occur, we can enumerate exactly how many copies of various constellations will survive each stage. In this paper, we study these systems of constellations of a fixed sum. Viewing them as discrete dynamic systems, we are able to characterize the populations of constellations for sums including the first few primorial numbers: 2, 6, 30. Since the eigenvectors of the discrete dynamic system are independent of the prime -- that is, independent of the stage of the sieve -- we can characterize the asymptotic behavior exactly. In this way we can give exact ratios of the occurrences of the gap 2 to the occurrences of other small gaps for all stages of Eratosthenes sieve.

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arXiv2013

Estimating constellations among primes - I. Uniformity

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression. For these initial estimates, we assume that the copies of a given constellation tend toward a uniform distribution in the cycle of gaps, as the recursion progresses. Our simple estimates based on the recursion of gaps and the assumption of uniformity appear to have correct asymptotic behavior, and they exhibit a systematic error correlated to length of the constellation.

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arXiv2013

Observations on the Perturbed Wedge

Santos' construction of the first known counterexample to the Hirsch conjecture, for bounded polytopes, follows the strategy of first finding a counterexample to the nonrevisiting conjecture. Santos constructs a $5$-dimensional all-but-simple spindle $(P,x,y)$ of length $6$, which is a counterexample to the nonrevisiting conjecture. For simple polytopes, if we had a counterexample to the nonrevisiting conjecture, we would produce the corresponding counterexample to the Hirsch conjecture through repeated wedging, over all the facets not incident to $x$ or $y$. However, Santos $5$-dimensional spindle is not simple. Every facet is incident to either $x$ or $y$, so we need an alternate method to produce the corresponding counterexample to the Hirsch conjecture. Santos has offered the perturbed wedge to accomplish this. In these working notes, we offer some technical details regarding the nonsimplicities under iterations of the perturbed wedge construction. NOTE: these are working notes about the construction.

math.CO↗
arXiv2013

Two Observations on the Perturbed Wedge

Francisco Santos has described a new construction, per- turbing apart a non-simple face, to offer a counterexample to the Hirsch Conjecture. We offer two observations about this perturbed wedge con- struction, regarding its effect on edge-paths. First, that an all-but- simple spindle of dimension d and length d + 1 is a counterexample to the nonrevisiting conjecture. Second, that there are conditions under which the perturbed wedge construction does not increase the diameter. NOTE: These are simply working notes, offering two observations on the construction identified by Santos.

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