cube triangular numbers and intersections of sequences

[Max A.]

On 12/30/06, Tanya Khovanova <tanyakh at tanyakhovanova.com> wrote:

> I can't find a cube triangular number other than 1. Is there a proof that they do not exist?

Suppose that n*(n+1)/2 = m^3. Then
(2*n+1)^2 - 1 = (2*m)^3
or
(2*n+1)^2 - (2*m)^3 = 1
According to the Catalan conjecture (proved recently) there is the
only pair of perfect powers differing in 1, which is 3^2 - 2^3 = 1.
Therefore, 2*n + 1 = 3  and 2*m = 2, implying n=m=1.

Max