This paper introduces one approach for a searching a prime number following the given prime number. The essence of the proposed approach is as follows. A sequence of two-element sets is constructed consisting of numbers enclosed between numbers that are multiples of three. The first set of the sequence contains the given prime number. Then the odd number of the next member of the introduced sequence is examined. Criteria for determining whether an odd number is composite or not are proposed. If it is composite, we pass to the study of the odd number of the next member of the sequence. If it is not composite, then it is the required prime number. The proposed approach is theoretically substantiated. In particular, two propositions are proved and algorithms for its implementation are elaborated.