Eccoci.

Let us define a geometric triangle as an integer sided triangle with sides $a \le b \le c$ so that its sides form a geometric progression, i.e. $b^2 = a \cdot c$

An example of such a geometric triangle is the triangle with sides $a = 144$, $b = 156$ and $c = 169$.

There are $861805$ geometric triangles with perimeter $\le 10^6$.

How many geometric triangles exist with perimeter $\le 2.5 \cdot 10^{13}$?