Are there finitely or infintely many prime solutions to $a^2 + b^2 = c^2 + 241$ such that $a^3 + b^3 + c^3$ is also prime?
Richard Borcherd's first lecture on algebraic geometry has, as an example, a birational correspondence between the unit circle and the y-axis that generates the Pythagorean triples.
It also has an algebraic characterization of Pythagorean triples.
The very first thing Borcherds did with $a^2 + b^2 = c^2$ is reduce it mod two, yielding:
$$ a^2 + b^2 \equiv_2 c^2 \implies \text{at least one of $a$ and $b$ is even if $\{a, b, c\}$ are coprime} $$
Borcherds uses this fact to characterize the Pythagorean triples algebraically, but reducing $a^2 + b^2 = c^2$ mod two can also be used, with a little extra reasoning, to prove that $a^2 + b^2 = c^2$ has no prime solutions.
This got me interested in prime solutions to a different equation, shown below:
$$ a^2 + b^2 = c^2 + 241 $$
A simple computer search suggests to me that there are infinitely many prime solutions to the above equation, although I don't have a proof of this fact.
However, solutions where $a^3 + b^3 + c^3$ is also prime are much less common.
The only one I have found so far is:
$$ 11^2 + 31^2 = 29^2 + 241 $$
Since $ 11^3 + 31^3 + 29^3 $ is $55511$, which is prime.
Are there finitely or infintely many prime solutions to $a^2 + b^2 = c^2 + 241$ such that $a^3 + b^3 + c^3$ is also prime?
Some solutions: $$ \matrix{ 31 & 11 & 29 \cr 113 & 79 & 137 \cr 179 & 31 & 181 \cr 479 & 109 & 491 \cr 691 & 421 & 809 \cr 1279 & 827 & 1523 \cr 3023 & 2389 & 3853 \cr 3089 & 1399 & 3391 \cr 4229 & 131 & 4231 \cr 4339 & 3491 & 5569 \cr }$$
The generalized Bunyakovsky conjecture implies that there are infinitely many solutions, but of course there's no proof.
For a bonus, here are two cases where the same number appears in two different solutions (not just by exchanging $p$ and $q$):
$$ \matrix{11551 & 3019 & 11939\cr 39227 & 3019 & 39343\cr 20509 & 20431 & 28949\cr 97861 & 20431 & 99971\cr} $$