Carl Pomerance

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math.NT2025

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

Carl Pomerance, Andreas Weingartner

A famous conjecture of Erd\H os and Straus is that for every integer $n \geq 2$ , $4/ n$ can be represented as $1/ x + 1/ y + 1/ z$ , where $x, y, z$ are positive integers. This conjecture was ge…

math.NT2025

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

Alina Ostafe, Carl Pomerance, Igor E. Shparlinski

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 arg…

math.NT2024

Shifted-prime divisors

Steve Fan, Carl Pomerance

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 influe…

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

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

Shifted-prime divisors

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