A cubic triangular number is a positive integer that is simultaneously cubic and triangular. Such a number must therefore
satisfy 
for some positive integers 
and 
, where 
is a triangular number,
so
 |
(1)
|
But then
 |
(2)
|
or
 |
(3)
|
According to Catalan's conjecture (now a theorem), the only pair of perfect powers differing by 1 is 
and 
, so the unique cubic triangular number has 
and 
, implying 
.
