Counting the number of positive prime solutions to third degree Thue equations equal a prime.

Question

Counting the number n of positive (x,y) = (p,q), p and q primes, solutions to third degree Thue equations equal a prime r, and p,q,r <1000 i get with the Pari gp built-in functions thueinit() and thue(), in a non exhaustive search:

n=9 for $3x^{3} -3x^{2}y +y^{3} =r $. They are, written in the form $(p,q,r)$ :

$(3,5,71);(2,5,89);(5,7,193);(5,2,233);(2,7,283);(7,5,419);(7,2,743);(7,11,743);(5,11,881)$

Any particular third degree Thue equation equal a prime with a number n of positive prime solutions less than 10^3; n >9 ?

I would be very surprised if anybody could find a n >13

Answer 1

$p^3-p^2q+q^3=r$ is satisfied by the $10$ prime triples $$(p,q,r)=(2,3,23),(2,5,113),(3,2,17),(3,5,107),(3,7,307),(5,2,83),(5,7,293),(7,3,223),(7,5,223),(11,7,827),$$ all with positive entries below $1000$.

Answer 2

This smacks somewhat of numerology, but I'll bite. Hows about $p^3-4p^2q+ p q^2+7q^3=r$, for $$ \begin{array}{c} (p,q,r)=(2,3,167), (2,5,853), (3,2,23), (3,5,797), (5,3,59), \\ (7,3,7), (11,2,463), (11,3,167), (11,5,61), (11,7,883), (13,2,953), \\ (13,5,17), (13,7,503), (17,5,433), (19,7,83), (23,7,883), (23,11,991), \\ (29,11,211), (29,13,937), (41,17, 853), (43,17,593), (47,19,919). \end{array} $$ This was found with a short search. One would expect to find examples with many more such values, if one was so inclined.....

P.S. Hi Gerry!