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

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

2009-07-12

arxiv.org/abs/0907.1940v4

Mathematics

Number Theory

This is a very preliminary draft, and maybe contains some mistakes

Scientific paper

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 $0\leq h\leq K\log x$.

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