Synchronization in a one-dimensional chain of Kuramoto oscillators with periodic boundary conditions is studied. An algorithm to rapidly calculate the critical coupling strength $$K_c$$ K c for complete frequency synchronization is presented according to the mathematical constraint conditions and the periodic boundary conditions. By this new algorithm, we have checked the relation between $$\langle K_c\rangle $$ 〈 K c 〉 and $$N$$ N , which is $$\langle K_c\rangle \sim \sqrt{N}$$ 〈 K c 〉 ∼ N , not only for small $$N$$ N , but also for large $$N$$ N . We also investigate the heavy-tailed distribution of $$K_c$$ K c for random intrinsic frequencies, which is obtained by showing that the synchronization problem is equivalent to a discretization of Brownian motion. This theoretical result was checked by generating a large sample of $$K_c$$ K c for large $$N$$ N from our algorithm to get the empirical density of $$K_c$$ K c . Finally, we derive the permutation for the maximum coupling strength and its exact expression, which grows linearly with $$N$$ N and would provide the theoretical support for engineering applications.