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Let   $F=x^2+y^2-z^2$  , and let   $x_0 \in \mathbb{Z}^3$   be a   $primitive$   solution to   $F(x_0)=0$  , e.g., so that its coordinates share no nontrivial divisor. Let   $\Gamma \leq \mathrm{SO_F(\mathbb{Z})}$   be a thin subgroup. We consider the resulting thin orbits of Pythagorean triples   $x_0 \cdot \Gamma$  —specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many   $R$  -almost primes in these three cases whenever   $\Gamma$   has exponent   $\delta_\Gamma&gt;\delta_0(R)$   for explicit   $R$  ,   $\delta_0$  .

Almost Primes in Thin Orbits of Pythagorean Triangles