Integral Brauer–Manin obstructions for sums of two squares and a power

摘要

We use Brauer–Manin obstructions to explain failures of the integral Hasse principle and strong approximation away from ∞ for the equation x~2 + y~2 + z~k = m with fixed integers k ≥ 3 and m. Under Schinzel’s hypothesis (H), we prove that Brauer–Manin obstructions corresponding to specific Azumaya algebras explain all failures of strong approximation away from ∞ at the variable z. Finally, we present an algorithm that, again under Schinzel’s hypothesis (H), finds out whether the equation has any integral solutions.