We prove new upper bounds on the number of representations of rational numbers $\frac{m}{n}$ as a sum of four 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 a 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 four unit fractions.

中文翻译：

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四个及更多单位分数的和以及近似参数化


我们证明了有理数表示数量的新上限 $\frac{m}{n}$ 作为四个单位分数的总和，给出五个不同的区域，具体取决于 $m$ 按照 $n$ 。特别是，我们改进了最相关的案例，当 $m$ 很小，并且当 $m$ 接近于 $n$ 。这些改进不仅源于研究解决方案集的完整参数化，还源于适当简化该解决方案集。所有参数的某些子集定义了所有解的集合，直到除数函数的应用，这对解的数量的上限影响很小。这些“近似参数化”是使计算机程序能够过滤大量方程和不等式的关键点。此外，这个结果导致有理数的表示数量作为四个以上单位分数之和的新上限。