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

Abstract

We use Brauer–Manin obstructions to explain failures of the integral Hasse principle and strong approximation away from ∞ for the equation x2+y2+zk=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.