TITLE: How many Pythagorean triples are there in which every member is triangular?
QUESTION [12 upvotes]: How many Pythagorean triples $(a,b,c)$ are there such that $a, b$ and $c$ are triangular?
Any two solutions with only $a$ and $b$ interchanged are considered equivalent.
The question of existence was posed by Zarankiewicz and answered by Sierpinski in Sur les nombres triangulaires carrés (1961):

Such a triangle is known whose sides are $t_{132} = 8778, t_{143} = 10296$ and $t_{164} = 13530$, but we do not know if there are others and if their number is finite.

A partial result: Since $t_n^2 = 1^3 + \ldots + n^3$, the equation $t_x^2 + t_y^2 = t_z^2$ is equivalent to $t_x^2$ being a sum of $k = z - y$ consecutive cubes. This leads one to consider elliptic curves.
In Pythagorean triples and triangular numbers (1979) by D. W. Ballew and R. C. Weger it is shown by applying a theorem of Siegel that for a given $k$ there are only finitely many solutions. Actually finding such solutions has been done by R. J. Stroeker: On the sum of consecutive cubes being a perfect square (1995), without the restriction that the square is some $t_x^2$.

REPLY [3 votes]: It was proved in http://arxiv.org/abs/0811.2477 (On certain diophantine equations related to triangular and tetrahedral numbers, by Maciej Ulas) that the equation
$t_x^2+t_y^2=t_z^2$ has inﬁnitely many solutions in rational numbers. It seems the number of integer solutions of this equation is still an open problem. It is also an open problem whether there exists a right triangle all sides of which are tetrahedral numbers of the form $T_n=\frac{1}{6}n(n+1)(n+2)$ (a tetrahedral version of the question asked).