Prime integers and their generalizations play important roles in protocols for secure transmission of information via open channels of telecommunication networks. Generation of multidigit large primes in the design stage of a cryptographic system is a formidable task. Fermat primality checking is one of the simplest of all tests. Unfortunately, there are composite integers (called Carmichael numbers) that are not detectable by the Fermat test. In this paper, the authors consider modular arithmetic based on complex integers and provide several tests that verify the primality of real integers. Although the new tests detect most Carmichael numbers, there are a small percentage of them that escape these tests.