In N-body problem, a simple choreography solution is a periodic solution in which all the masses chase each other on a single loop. The most famous solution of this kind is the Figure-Eight solution found by Chenciner and Montgomery in 2000.
In this talk we will show for the Newtonian N-body problem, $N >= 3$, with equal masses, there are at least $2^{N-3} + 2^{[(N-3)/2]}$ different main simple choreography solutions. As a result this confirms a conjecture given by Chenciner and etc. in 2002.