[seqfan] Pythagorean triples of triangular numbers?

[jonscho at hiwaay.net]

Dear Sequence Fans:

During some recent editing of A213188, Andrew Howroyd added a comment  
that there is only one known example of a Pythagorean triple  
consisting of three triangular numbers: 8778^2 + 10296^2 = 13530^2  
(i.e., T(132)^2 + T(143)^2 = T(164)^2).

I did a very simplistic brute force search and determined that there  
aren't any other examples in which the hypotenuse is less than  
2*10^10, and asked him if he knew how far the search had been taken.   
He did a Google search for the numbers in the only known example and  
found a few hits, one of which said that if there's another example,  
its hypotenuse is > 17680431, but he commented that this seemed  
absurdly weak.  He added, "Part of me thinks this ought to be a well  
known unsolved problem, and because of the numerous constraints  
someone would have found a smarter way to search using elliptic curves  
etc so my guess is that you are not even close. (but don't give up!).  
On the other hand, that paper that briefly mentioned this fact was  
written in the 70's and the lack of google hits means perhaps it has  
been overlooked (or there is no way to solve with elliptic curves)."

I asked whether this might be a question suitable for the Seqfan  
mailing list, and Neil asked if one of us could submit it.

Does anyone know whether --

-- any other solutions have been found?

-- anyone has proved that there are no other solutions?

-- any lower bounds hugely larger than 2*10^10 for the hypotenuse (or  
for the other sides) have been established for any possible solution  
other than (8778, 10296, 13530)?

-- elliptic curves offer a smart way to search for solutions?

Thanks!

-- Jon