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, if p = 3, then 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 algorithms for testing primality. They are also used in generating large random numbers and in cryptography. The search for new Mersenne primes continues, with many of the largest known primes being of this form.