Title: The Burnside problem for odd exponents

Abstract: We show that the free Burnside groups $B(n,m)$ are infinite for $n\geq 557$ and $m\geq 2$. The proof uses iterated small cancellation theory where the induction based on the nesting depth of relators. The main instrument at every step is a new concept of a certification sequence. This decreases the best known lower bound in the Burnside problem for odd exponents from 665 to 557.