A Mersenne number is a specific type of integer that is one less than a power of two. It is expressed in the form M_n = 2^n - 1 , where n is a positive integer. For example, when n = 3 , the Mersenne number is 2^3 - 1 = 7 . These numbers are named after the French monk Marin Mersenne, who studied their properties in the early 17th century. Mersenne numbers are particularly interesting in the field of number theory and are closely related to Mersenne primes. A Mersenne prime is a Mersenne number that is also a prime number. For instance, M_2 = 3 and M_3 = 7 are both Mersenne primes. These numbers have applications in computer science, especially in cryptography and {random number