# Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines

## TL;DR

This paper investigates the Brauer-Manin obstruction to the existence of integer solutions for specific cubic surface equations, showing that under certain conditions, no such obstruction prevents solutions.

## Contribution

It provides new results on the absence of Brauer-Manin obstructions for particular families of affine cubic surfaces related to sums of cubes.

## Key findings

- No Brauer-Manin obstruction when a ≠ 4,5 mod 9 for x^3+y^3+z^3=a
- No Brauer-Manin obstruction for x^3+y^3+2z^3=a under the same conditions
- Results clarify when integer solutions are not obstructed by Brauer-Manin

## Abstract

Let a be a nonzero integer. If a is not congruent to 4 or 5 modulo 9 then there is no Brauer-Manin obstruction to the existence of integers x, y, z such that x^3+y^3+z^3=a. In addition, there is no Brauer-Manin obstruction to the existence of integers x, y, z such that x^3+y^3+2z^3=a.

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Source: https://tomesphere.com/paper/0911.3539