What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

Solution 1:

Question 1 was answered by Peter Kosinar in the comments, and a general version of Question 2 was answered by Michael Stoll in this MO post.

The general case of Question 3 is in this post.

P.S. One nice thing about these cubics is that, starting with Ramanujan's general cubic identity, they are a special case, yielding the simple,

$$(a+b\,x_1)^{1/3}+(a+b\,x_2)^{1/3}+(a+b\,x_3)^{1/3}=\big(c+\sqrt[3]{dp}\big)^{1/3}$$

for some rational $a,b,c,d$. For example, using $p=109$, so $x^3 + x^2 - 36x - 4=0$, then,

$$(2+x_1)^{1/3}+(2+x_2)^{1/3}+(2+x_3)^{1/3}=\big({-19}+\sqrt[3]{4\cdot109}\big)^{1/3}=1.553389\dots$$