A concise proof of the 8 case of Sylvester's conjecture

@misc{yin2026concise,
  title = {{A concise proof of the 8 case of Sylvester's conjecture}},
  author = {Hongbo Yin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08015},
  eprint = {2610.08015},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08015},
  note = {CC BY-NC-SA 4.0},
}

TY  - UNPB
TI  - A concise proof of the 8 case of Sylvester's conjecture
AU  - Hongbo Yin
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.08015
UR  - https://arxiv.org/abs/2610.08015
AB  - The 150 years old Sylvester's conjecture asserts that every prime p≡ 4,7,8mod 9 is the sum of two rational cubes. Recently, a proof of this old problem is obtained by the work of mine for the case 4,7 \cite{Yin} and the work of Burungale-Tian for the case 8 \cite{BT}. In this paper we give a shorter proof by deep mining my result in \cite{Yin} combining with the ideas of my previous work on case 8 in \cite{Yin8}. In particular, we use the ordinary trace and the Kronecker congruence of modular functions, but do not use the √(−3)-division boundary and Fermat isogeny as in \cite{BT}.
ER  -