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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.

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Mirko
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  • Try $i=4$, $j=5$, and $t=15$. –  Dec 05 '15 at 16:45
  • @ByronSchmuland hey, it was not sup[posed to be that elementary :) so where did I go wrong with my other answer? – Mirko Dec 05 '15 at 16:51
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    Sorry but the word "impossible" is like a red flag to me. ;) The first thing I do is to run a small program to look for counterexamples. I haven't looked carefully at the other answer yet. –  Dec 05 '15 at 16:54
  • Is the argument in the other answer reversible? If not, there is no contradiction here. – rogerl Dec 05 '15 at 17:03

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