# Central configurations in planar $n$-body problem with equal masses for $n = 5, 6, 7$

## TL;DR

This paper provides a computer-assisted classification of all central configurations in the planar Newtonian $n$-body problem with equal masses for $n=5,6,7$, revealing symmetry properties and existence of asymmetric configurations for larger $n$.

## Contribution

It offers the first complete listing of central configurations for $n=5,6,7$ with equal masses, including symmetry analysis and new existence results for $n=8,9,10$.

## Key findings

- All configurations for $n=5,6,7$ have reflective symmetry.
- Existence of asymmetric configurations for $n=8,9,10$.
- Computer-assisted proof ensures completeness of classification.

## Abstract

We give a computer assisted proof of the full listing of central configuration for $n$-body problem for Newtonian potential on the plane for $n=5,6,7$ with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For $n=8,9,10$ we establish the existence of central configurations without any reflectional symmetry.