论文标题
连续三个近方形的无平方数字
Three consecutive near-square squarefree numbers
论文作者
论文摘要
在本说明中，我们通过使用T. Estermann的和S. Dimitrov的论点来证明，并且具有基本不平等的论点，即所有的$ n $都有无限的$ n $ $ n^2+1，n^2+1，n^2+2 $和$ n^2+3 $是SquareFree。在两个连续数字相同形式的情况下，我们还稍微改善了误差项，以便我们能够证明以下渐近公式。 \ begin {align*} \ sum_ {n \ le X}μ^2（n^2+1）μ^2（n^2+2）μ^2（n^2+3）\ sim \ dfrac {7} {18} {18} \ prod_ {p> 3} \ left（1- \ dfrac {1- \ dfrac {3+ frac {-1} {p} \ right）+\左（\ frac {-2} {p} {p} \ right）+\ left（\ frac {-3} {-3} {p} {p} \ right）} {p^2} \ right）x。 \ end {align*}
In this note, we prove by using T. Estermann's and S. Dimitrov's arguments with an elementary inequality that there are infinitely many $n$ for which all of the numbers $n^2+1,n^2+2$ and $n^2+3$ are squarefree. We also improve the error term slightly in the case of two consecutive numbers of the same form, so that we are able to prove the following asymptotic formula. \begin{align*} \sum_{n\le X}μ^2(n^2+1)μ^2(n^2+2)μ^2(n^2+3)\sim\dfrac{7}{18}\prod_{p>3}\left(1-\dfrac{3+\left(\frac{-1}{p}\right)+\left(\frac{-2}{p}\right)+\left(\frac{-3}{p}\right)}{p^2}\right)X. \end{align*}
