Show that there are no positive integers $t,i,j$ with $j>i$ such that:
$\displaystyle \frac{t(t+1)}2=\frac{2i(j-i)j(j+i)}3$
If possible provide an elementary proof.
I believe the statement is correct as implied by the following question,
combined with @WillJagy 's answer and my answer there.
But that question is also looking for a more elementary answer.
Note that both sides of the equation in the present problem are integers, indeed $\displaystyle\frac{t(t+1)}2$ is a triangular number, while if $3\not|\ i$ then $3|(j-i)j(j+i)$.
Edit. As shown by @ByronSchmuland in the comments, $t=15$, $j=5$, $i=4$ satisfy the above equation. Something must not be right with my other answer.