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 
.
