For the ternary quadratic form Q(x) = x ~2 + y ~2 - z ~2 and a non-zero Pythagorean triple x _0 ∈ ? ~3 lying on the cone Q(x) = 0, we consider an orbit Q = x _0Γ of a finitely generated subgroup Γ < SO _Q(?) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in Q whose hypotenuse, area, and product of side lengths have few prime factors, where "few" is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants.
展开▼
