Mathematical Theory and Applications ›› 2024, Vol. 44 ›› Issue (2): 65-79.doi: 10.3969/j.issn.1006-8074.2024.02.005
Previous Articles Next Articles
You Lihua, Li Jiayin, Yuan Pingzhi*
Online:
Published:
Supported by:
Abstract:
In this paper, we study the (distinct) positive integer solution of the equation
\begin{equation*}\label{eq12}\frac{k}{n} = \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_t}\end{equation*} with $n>k\geq 2$ and $ t\geq 2$.
We show that the above equation has at least one distinct positive integer solution if it has a positive integer solution.
When $k=5$, we show the above equation has at least one distinct positive integer solution for all $n\geq 3$
except possibly when $n\equiv 1, 5041, 6301, 8821, 13861, 15121(\mbox{mod } 16380)$ with $t=3$,
and for all $n\geq 3$ except possibly when $n\equiv 1, 81901(\mbox{mod } 163800)$ with $t=4$.
Furthermore, we point out that the above equation has at least one distinct positive integer solution for all $n(>k)$
when $t\geq k\geq 2$.
Key words: Diophantine equation, Positive integer solution, Distinct, Erd\"{o}s-Straus conjecture
You Lihua, Li Jiayin, Yuan Pingzhi. Existence of Distinct Positive Integer Solutions to a Generalized Form of Erdös-Straus Conjecture[J]. Mathematical Theory and Applications, 2024, 44(2): 65-79.
0 / / Recommend
Add to citation manager EndNote|Ris|BibTeX
URL: http://mta.csu.edu.cn/EN/10.3969/j.issn.1006-8074.2024.02.005
http://mta.csu.edu.cn/EN/Y2024/V44/I2/65