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

Source abstract

The 150 years old Sylvester's conjecture asserts that every prime $p\equiv 4,7,8\mod 9$ 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}. Burungale-Tian's paper essentially uses the results of paper \cite{Yin}, but does not make full use of the power. Their proof for the case $8$ is complicated and hard to follow. In this paper we give a much simpler and direct 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 Gross-Zagier formula, $\sqrt{-3}$-division boundary and Fermat isogeny as in \cite{BT}.This concise proof provides a perfect end to this old concise conjecture.