A variety of Euler's sum of powers conjecture

Cai, Tianxin, and Zhang, Yong. "A variety of Euler's sum of powers conjecture." Czechoslovak Mathematical Journal 71.4 (2021): 1099-1113. <http://eudml.org/doc/297828>.

@article{Cai2021,
abstract = {We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system \[ \{\left\lbrace \begin\{array\}\{ll\} n=a\_\{1\}+a\_\{2\}+\cdots +a\_\{s-1\},\\ a\_\{1\}a\_\{2\}\cdots a\_\{s-1\}(a\_\{1\}+a\_\{2\}+\cdots +a\_\{s-1\})=b^\{s\} \end\{array\}\right.\} \] has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ 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 $s\ge 4$. As a corollary, for $s\ge 4$ 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 $s\ge 4$ and a fixed positive integer $n$.},
author = {Cai, Tianxin, Zhang, Yong},
journal = {Czechoslovak Mathematical Journal},
keywords = {Euler's sum of powers conjecture; elliptic curve; positive integer solution; positive rational solution},
language = {eng},
number = {4},
pages = {1099-1113},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A variety of Euler's sum of powers conjecture},
url = {http://eudml.org/doc/297828},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Cai, Tianxin
AU - Zhang, Yong
TI - A variety of Euler's sum of powers conjecture
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 4
SP - 1099
EP - 1113
AB - We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system \[ {\left\lbrace \begin{array}{ll} n=a_{1}+a_{2}+\cdots +a_{s-1},\\ a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots +a_{s-1})=b^{s} \end{array}\right.} \] has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ 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 $s\ge 4$. As a corollary, for $s\ge 4$ 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 $s\ge 4$ and a fixed positive integer $n$.
LA - eng
KW - Euler's sum of powers conjecture; elliptic curve; positive integer solution; positive rational solution
UR - http://eudml.org/doc/297828
ER -