What is the pattern for finding prime numbers?

Could you please elaborate on the methodology employed to identify prime numbers? Is there a discernible pattern or set of rules that consistently lead to the discovery of these special numbers? Are there any mathematical algorithms or properties that facilitate the search process? Additionally, how do mathematicians ensure that they have thoroughly searched through all possible candidates and not overlooked any potential primes? Lastly, are there any limitations or challenges associated with identifying prime numbers, particularly as the numbers grow larger and more complex?