I recently discovered a proof of the following. Let $p$ be a prime that's $1 \bmod {3}$. Suppose that $p$ is not represented by the principal quadratic form $(1,9,81)$ of discriminant $-243$ (The first three $p$ that are represented by this form are $61$, $67$, and $73$). Then neither $3p$ nor $3p^2$ is a sum of two rational cubes.


The proof is elementary (at the undergraduate level) in the following sense. In the statement of the theorem I impose two further conditions on $p$, each of which like the non-representability by $(1,9,81)$ is easily checked for any given $p$. The first of these supplementary conditions in fact holds for all $p$, while the second (that $3$ is not a cube in $\mathbb{Z}/p\mathbb{Z}$) is equivalent to $p$ not being representable by $(1,9,81)$. But the assertions of this last sentence are cubic reciprocity results authored by Eisenstein and Gauss.


**Question 1**


In view of this last paragraph, my result could have been proved even as early as the time of Gauss. But I think it may be new - I've seen no reference in Sylvester. Does anyone have a historical reference?


**Question 2**


When $p$ is $61$, $67$ or $73$, then $3p$ is indeed the sum of two rational cubes. So it's natural to ask whether the converse to my theorem holds. When $p$ is represented by $(1,9,81)$ is it true that $3p$ and $3p^2$ are sums of two rational cubes? This has something of the flavor of Sylvester's question about representing primes congruent to $4$, $7$ or $8 \bmod{9}$ (and their squares) as sums of two rational cubes. Before the work of Elkies, Dasgupta-Voight and Kriz which (I hope) have settled the question there was $3$-descent work by Selmer that "reduced" things to the parity conjecture. What does the approach of Selmer reveal here? And has anyone looked at the elliptic curves explicitly, and tried to use the Heegner point approach to produce solutions to the equations?



Writing $3p$ when $p \equiv 1 \pmod{3}$ as a sum of two rational cubes. Is this result new? And what about its converse?
For question 1.


The condition $p$ is represented by $(1,9,81)$ or equivalently by $(1,1,61)$ is equivalent to the condition that $p\equiv 1\mod 3$ and $3 \mod p$ is not a cube (this is exercise 9.10 in Cox's book 'Primes of the forms...').
And Satge's paper 'Groupes de Selmer et corps cubiques' already proved under this condition, neither $3p$ nor $3p^2$ is a sum of two rational cubes.


For question 2.


I checked using Sagemath that for $p=61,67,73,103,151,193,271$ the rank of the related elliptic curve of $3p$ is 2 while for $p=307$ the rank is 0. So the converse is not true. Similar thing also occurs for $3p^2$. However for $p\equiv 2,5\mod 9$, it is proved that $3p$ and $3p^2$ are both sums of two cubes by constructing the Heegner point in the paper <https://link.springer.com/content/pdf/10.1007/s00208-022-02370-3.pdf>


