A Mersenne prime is a special type of prime number that can be expressed in the form 2^p - 1, where p is also a prime number. For example, when p = 3, the calculation gives 2^3 - 1 = 7, which is a prime number. Mersenne primes are named after the French monk Marin Mersenne, who studied these numbers in the early 17th century. Mersenne primes are significant in number theory and have applications in computer science, particularly in cryptography and random number generation. They are also used in the search for large prime numbers, with the largest known prime number as of now being a Mersenne prime.