Almost Primes in Thin Orbits of Pythagorean Triangles

TL;DR

This paper investigates the distribution of almost prime hypotenuses, areas, and coordinate products within thin orbits of Pythagorean triples generated by a specific thin subgroup, establishing conditions for infinitely many such almost primes.

Contribution

It introduces new results on the occurrence of almost primes in thin orbits of Pythagorean triples, with explicit bounds related to the subgroup's exponent.

Findings

Infinitely many R-almost primes in hypotenuses, areas, and coordinate products.

Explicit conditions on subgroup exponent for the existence of these almost primes.

Extension of classical prime distribution results to thin orbit settings.

Abstract

Let $F=x^2+y^2-z^2$, $x_0 \in \mathbb{Z}^3$ primitive with $F(x_0)=0$, and $\Gamma \leq SO_F(\mathbb{Z})$ be a finitely generated 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 $\delta_\Gamma>\delta_0(R)$ for explicit $R$, $\delta_0$.