A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.


This is the actual question which I'm trying to solve:-


For a research problem that I'm working on, I need to solve the following system of Diophantine equations:-


$ a^3 + 40033 = d$,
$ b^3 + 39312 = d$,
$ c^3 + 4104 = d$, where $a,b,c,d>0$ are all DISTINCT positive integers and $a,b,c∉${$2,9,15,16,33,34$}.


Sorry about the earlier mishap. Any and all help is appreciated! Thanks!



Help with this system of Diophantine equations
The first two equations amount to $b^3 - a^3 = 721$. 


Now since for any solution $b^3 - a^3 \ge (a+1)^3 - a^3 = 3a^2 + 3a +1$ this directly gives an upper bound on $a$ namely $15$.


Now checking for which of $a=1, \dots, 15$ one has that $a^3 + 721$ is the third power of an integer, by calculating its third root for example one finds that this is only the case for $a=2$ and $a=15$ (where $b$ would be $9$ and $16$, respectively). 


Both are excluded, so already the first two equations have no solution. 


