\ heIn this paper is offered and theoretically is based the algorithm permissive with the lp of small number of arithmetic operations with arbitrary positive integer(N) to answer a question : is N composite or prime? The algorithm has a high operational speed which depends a little on value N ,and is based on The method of structurization of a set of positive integers (Np) developed by the author. In limits of a framework of this method is defined a special set of the structured integers (Ns) in which it becomes possibility for testing of any structured integers (Sn) on a membership of a set of composite structured integers (Nsc). Between by Np and Ns is established one-to-one correspondence : composite structured integers (Snc) are corresponded to composite positive integers . Prime structured integers are corresponded to prime positive integers Thus for testing arbitrary (N) it is necessary to map it into . Then we test obtained on a membership of If Sn is a member of then the output follows that tested N is also composite.If Sn is not a member of Nsc then the output follows that tested $N$ is also prime , since if Sn is not composite then it is prime ,tertiary is not given.