TITLE: Solve $2p+6=n^3$
QUESTION [6 upvotes]: How can I prove that if $p$ is a prime, there is an unique solution for the equation
$$2p+6=n^3$$
where $n$ is an integer?
I know that $p=29$ is the solution, but I don't know how to demonstrate that is the only one. Anyone can help me?

REPLY [7 votes]: Motivated by Arentino's observation, here are some other solutions as provided by Mathematica 10 (all solutions for $n\le 1000$). They are listed in the form $\{n,p\}$.
Here's the code:
Do[If[PrimeQ[(n^3 - 6)/2], Print[{n, (n^3 - 6)/2}]], {n, 2 Range[500]}]

{4,29}
{28,10973}
{32,16381}
{44,42589}
{50,62497}
{58,97553}
{98,470593}
{128,1048573}
{148,1620893}
{154,1826129}
{172,2544221}
{182,3014281}
{224,5619709}
{242,7086241}
{254,8193529}
{268,9624413}
{274,10285409}
{280,10975997}
{302,13771801}
{322,16693121}
{358,22941353}
{394,30581489}
{428,39201373}
{442,43175441}
{452,46172701}
{508,65548253}
{518,69495913}
{568,91625213}
{574,94559609}
{592,103737341}
{604,110174429}
{610,113490497}
{620,119163997}
{694,167127689}
{722,188183521}
{728,192914173}
{730,194508497}
{760,219487997}
{764,222971869}
{784,240945149}
{788,244651933}
{802,257924801}
{812,267693661}
{814,269676569}
{830,285893497}
{938,412646833}
{994,491053889}