Almost Primes in Thin Orbits of Pythagorean Triangles

2016/05/17 by Ehrman, Max
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1605.05265

Abstract

Let F=x²+y²-z², x₀ ∈ ℤ³ primitive with F(x₀)=0, and Γ≤ SO_F(ℤ) be a finitely generated thin subgroup. We consider the resulting thin orbits of Pythagorean triples x₀ ⋅ Γ - specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many R-almost primes in these three cases whenever Γ has exponent δ_Γ>δ₀(R) for explicit R, δ₀.

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