On the number of distinct prime factors of $nj+a^hk$

TL;DR

This paper demonstrates that for large x, a significant proportion of integers within a specific interval have a high number of distinct prime factors for various algebraic forms, extending understanding of prime factorization distribution.

Contribution

It establishes lower bounds on the number of integers with many distinct prime factors in certain algebraic progressions, advancing previous results on prime factorization patterns.

Findings

At least x^{1-ε} integers in the interval have many prime factors.

The number of prime factors exceeds a logarithmic triple iterated growth rate.

Results hold uniformly for a wide range of parameters and algebraic forms.

Abstract

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Then for any given $K\geq 2$, small $\epsilon>0$ and sufficiently large (only depending on $K$ and $\epsilon$) $x$, there exist at least $x^{1-\epsilon}$ integers $n\in[x,(1+K^{-1})x]$ such that $\omega(nj\pm a^hk)\geq(\log\log\log x)^{{1/3}-\epsilon}$ for all $2\leq a\leq K$, $1\leq j,k\leq K$ and $0\leq h\leq K\log x$.