Arithmetic properties of blocks of consecutive integers

TL;DR

This paper surveys the arithmetic properties of consecutive integer blocks, examining prime factors and powerfree parts, and shows the abc-conjecture implies the Erdős-Woods conjecture for k>2.

Contribution

It provides a comprehensive survey of known results and demonstrates that the explicit abc-conjecture implies the Erdős-Woods conjecture for all k>2.

Findings

Survey of prime factor properties in consecutive integers

Conditional proof linking abc-conjecture to Erdős-Woods conjecture

Results without assumptions on prime factor distributions

Abstract

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erd\H{o}s-Woods conjecture for each k>2.