Cubic Triangular Number

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 .

REFERENCES:

Alekseyev, M. "Re: Cube Triangular Numbers and Intersections of Sequences." seqfan@ext.jussieu.fr posting, 30 Dec 2006.

Spies, J. "Problem A NAW 5/5 nr. 4." Dec. 4, 2004. https://www.jaapspies.nl/mathfiles/problem2004-4A.pdf.