论文标题
简短间隔的功能数字的注释
A note on powerful numbers in short intervals
论文作者
论文摘要
在本说明中，我们有兴趣在短期内获得均匀的上限$（x，x，x + y] $。我们获得无条件的上限上限$ O（\ frac {y} {y log y}）$和$ o（y^{y^{11/12}）$ for所有强大的数字和$ y^y^$ y^1/2 $ - $ -2/2} $ - 我们证明了$ o（\ frac {y} {\ log^{1 +ε} y}）$用于平方的数字，而$ k $ -full数字则与$ k $ -full数字进行$ k \ ge 3 $相关的$ k $ -full数字。
In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and $y^{1/2}$-smooth powerful numbers respectively. Conditional on the $abc$-conjecture, we prove the bound $O(\frac{y}{\log^{1+ε} y})$ for squarefull numbers and the bound $O(y^{(2 + ε)/k})$ for $k$-full numbers when $k \ge 3$. They are related to Roth's theorem on arithmetic progressions and the conjecture on non-existence of three consecutive squarefull numbers.
