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Carl Pomerance

Publications and source records attributed to Carl Pomerance.

At least 19 recordsLinked to original sources
arXiv2026

Lines in the prime number graph

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line. Recently Sloane conjectured that $L(n) = O(n/\log n)$. We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all large $n$. Under RH we show that for large $n$ we have $B(n)=O(n^{3/4}(\log n)^{1/2})$ and $ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant $c'>0.$

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arXiv2026

Matchable numbers

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has an asymptotic density, which we compute, and we show that every squarefree number is matchable. We also present some related unsolved problems.

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arXiv2025

Exceptions to the Erd\H os--Straus--Schinzel conjecture

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge20$. A result of Vaughan is that for each $m$, most $n$'s have $m/n$ representable; we make the dependence on $m$ in this result explicit. In addition, we prove a result generalizing the problem to the sum of $j$ unit fractions.

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arXiv2025

Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros

We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a classical idea of P. Erd\H{o}s (1935). We then use similar ideas to get a tight upper bound on the number of linear recurrence sequences which attain a zero value.

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arXiv2024

Cyclotomic primes

Mersenne primes and Fermat primes may be thought of as primes of the form $\Phi_m(2)$, where $\Phi_m(x)$ is the $m$th cyclotomic polynomial. This paper discusses the more general problem of primes and composites of this form.

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arXiv2024

Shifted-prime divisors

Let $ω^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of $ω^*(n)$ are still somewhat a mystery. This paper addresses these higher moments and considers other related problems.

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arXiv2022

Long gaps in sieved sets

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (\log x)^δ $ where $δ>0$ depends only on the density of primes for which $I_p\ne \emptyset$. This improves on the "trivial" bound of $\gg x$. As a consequence, for any non-constant polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient, the set $\{ n \leq X: f(n) \hbox{ composite}\}$ contains an interval of consecutive integers of length $\ge (\log X) (\log\log X)^δ$ for sufficiently large $X$, where $δ>0$ depends only on the degree of $f$.

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arXiv2022

Permutations with arithmetic constraints

Let $S_{\rm lcm}(n)$ denote the set of permutations $π$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,π(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $π$ of $[n]$ such that $j\midπ(j)$ or $π(j)\mid j$ for each $j\in[n]$. Clearly $S_{\rm div}(n)\subset S_{\rm lcm}(n)$. We get upper and lower bounds for the counts of these sets, showing they grow geometrically. We also prove a conjecture from a recent paper on the number of "anti-coprime" permutations of $[n]$, meaning that each $\gcd(j,π(j))>1$ except when $j=1$.

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arXiv2022

Coprime permutations

Let $C(n)$ denote the number of permutations $σ$ of $[n]=\{1,2,\dots,n\}$ such that $\gcd(j,σ(j))=1$ for each $j\in[n]$. We prove that for $n$ sufficiently large, $n!/3.73^n < C(n) < n!/2.5^n$.

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arXiv2021

Coprime matchings

We prove that there is a matching between 2 intervals of positive integers of the same even length, with corresponding pairs coprime, provided the intervals are in $[n]$ and their lengths are $>c(\log n)^2$, for a positive constant $c$. This improves on a recent result of Bohman and Peng. As in their paper, the result has an application to the lonely runner conjecture.

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arXiv2021

The denominators of the Bernoulli numbers

We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct Bernoulli denominators and some related problems.

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arXiv2021

On a nonintegrality conjecture

It is conjectured that the sum $$ S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k} $$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$ such that $S_r(n)\in {\mathbb Z}$ for some $r\ge 1$ has asymptotic density $0$. Our principal tools are some deep results on the distribution of primes in short intervals.

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arXiv2021

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 Erdős, 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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arXiv2021

Some thoughts on pseudoprimes

We consider several problems about pseudoprimes. First, we look at the issue of their distribution in residue classes. There is a literature on this topic in the case that the residue class is coprime to the modulus. Here we provide some robust statistics in both these cases and the general case. In particular we tabulate all even pseudoprimes to $10^{16}$. Second, we prove a recent conjecture of Ordowski: the set of integers $n$ which are a pseudoprime to some base which is a proper divisor of $n$ has an asymptotic density.

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arXiv2021

A note on Carmichael numbers in residue classes

Improving on some recent results of Matomäki and of Wright, we show that the number of Carmichael numbers to $X$ in a coprime residue class exceeds $X^{1/(6\log\log\log X)}$ for all sufficiently large $X$ depending on the modulus of the residue class.

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arXiv2020

On the critical exponent for $k$-primitive sets

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-λ}$ over a primitive set is maximized by the primes if and only if $λ$ is at least the critical exponent $τ_1 \approx 1.14$. A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. One may similarly consider the critical exponent $τ_k$ for which the primes are maximal among $k$-primitive sets. In recent work the authors showed that $τ_2 < 0.8$, which directly implies the Erdős conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that $τ_k$ tends to zero as $k\to\infty$.

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arXiv2020

On primes and practical numbers

A number $n$ is practical if every integer in $[1,n]$ can be expressed as a subset sum of the positive divisors of $n$. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and Weingartner. In addition, essentially proving a conjecture of Margenstern, we show that all large odd numbers are the sum of a prime and a practical number. We also consider an analogue of the prime $k$-tuples conjecture for practical numbers, proving the "correct" upper bound, and for pairs, improving on a lower bound of Melfi.

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