论文标题
四个和更多单位分数和近似参数的总和
Sums of four and more unit fractions and approximate parametrizations
论文作者
论文摘要
我们证明，有理数$ \ frac {m} {n} $的代表数量的新上限是$ 4 $单位分数的总和，可提供五个不同的区域，具体取决于$ n $的$ m $。特别是，当$ m $很小时，当$ m $接近$ n $时，我们会改善最相关的情况。这些改进不仅源于研究解决方案集的完整参数化，而且还适当地简化了该集合。所有参数的某些子集定义了所有解决方案的集合，直到除数功能的应用，这对解决方案数量的上限几乎没有影响。这些“近似参数化”是使计算机程序能够通过大量方程和不平等的过滤的关键点。此外，该结果为理性数字的表示形式数量提供了新的上限，总和超过$ 4 $单位分数。
We prove new upper bounds on the number of representations of rational numbers $\frac{m}{n}$ as a sum of $4$ unit fractions, giving five different regions, depending on the size of $m$ in terms of $n$. In particular, we improve the most relevant cases, when $m$ is small, and when $m$ is close to $n$. The improvements stem from not only studying complete parametrizations of the set of solutions, but simplifying this set appropriately. Certain subsets of all parameters define the set of all solutions, up to applications of divisor functions, which has little impact on the upper bound of the number of solutions. These "approximate parametrizations" were the key point to enable computer programmes to filter through large number of equations and inequalities. Furthermore, this result leads to new upper bounds for the number of representations of rational numbers as sums of more than $4$ unit fractions.
