Is there any formula to find prime numbers?

Excuse me, could you elaborate on whether there exists a definitive formula or algorithm that reliably identifies prime numbers without the need for extensive trial and error? Is it possible to devise a mathematical equation that, given any integer, instantly determines whether it's prime or composite? If so, could you provide an example or outline of such a formula, and discuss any limitations or exceptions to its application? Additionally, how do current methods for finding prime numbers, such as the Sieve of Eratosthenes, compare in terms of efficiency and practicality to a hypothetical formula, if one were to exist?