A variety of Euler's sum of powers conjecture

Tianxin Cai; Yong Zhang

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 4, page 1099-1113
  • ISSN: 0011-4642

Abstract

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We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system n = a 1 + a 2 + + a s - 1 , a 1 a 2 a s - 1 ( a 1 + a 2 + + a s - 1 ) = b s has positive integer or rational solutions n , b , a i , i = 1 , 2 , , s - 1 , s 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 4 . As a corollary, for s 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 4 and a fixed positive integer n .

How to cite

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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 -

References

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