The expression "2^p - 1" represents a mathematical formula where "p" is a positive integer. This formula is significant in number theory, particularly in the study of Mersenne primes. A Mersenne prime is a prime number that can be expressed in the form of "2^p - 1" for some integer "p". For example, if "p" equals 3, then "2^3 - 1" equals 7, which is a prime number. The search for Mersenne primes has been a long-standing interest in mathematics, as these numbers have unique properties and applications in fields like cryptography and computer science.