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A prime number which can be written in the form 2ⁿ−1 is called a
Mersenne prime. The Mersenne number M_n is prime for many values of n,
for instance n=2, 3, 5, 7, 13.
It is necessary for n to be prime, but not every prime number n leads to a Mersenne prime 2ⁿ−1; for instance n=11 does
not give a Mersenne prime. The first few are:
2²−1 = 3,
2³−1 = 7,
2⁵−1 = 31,
2⁷−1 = 127,
2¹³−1 = 8191,
If you are interested in knowing the latest Mersenne Prime that has been found, visit The Great Internet Mersenne Prime Search.
Note that if 2ⁿ−1 is prime, then 2^p−1(2^p−1) is a perfect number.