| Title:
|
A variety of Euler's sum of powers conjecture (English) |
| Author:
|
Cai, Tianxin |
| Author:
|
Zhang, Yong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
1099-1113 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider a variety of Euler's sum of powers conjecture, i.e., whether the Diophantine system $$ \begin{cases} n=a_{1}+a_{2}+\cdots +a_{s-1},\\ a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots +a_{s-1})=b^{s} \end{cases} $$ has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\geq 3.$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s=3$, but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s\geq 4$. As a corollary, for $s\geq 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that it has infinitely many positive rational solutions for $s\geq 4$ and a fixed positive integer $n$. (English) |
| Keyword:
|
Euler's sum of powers conjecture |
| Keyword:
|
elliptic curve |
| Keyword:
|
positive integer solution |
| Keyword:
|
positive rational solution |
| MSC:
|
11D41 |
| MSC:
|
11D72 |
| MSC:
|
11G05 |
| idZBL:
|
Zbl 07442476 |
| idMR:
|
MR4339113 |
| DOI:
|
10.21136/CMJ.2021.0210-20 |
| . |
| Date available:
|
2021-11-08T16:01:56Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149240 |
| . |
| Reference:
|
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| Reference:
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