Almost prime Pythagorean triples in thin orbits

Details Almost prime Pythagorean triples in thin orbits Almost prime Pythagorean triples in thin orbits

2010-01-03

arxiv.org/abs/1001.0370v1

Mathematics

Number Theory

37 pages, 8 figures, 1 table

Scientific paper

For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in O 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.

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