Mersenne Primes Under the Microscope

What are Mersenne Primes?

The recent discovery of the largest known prime number which is 17 million digits highlighted the importance of Mersenne primes.  This newest prime number and the thirteen largest primes are all Mersenne primes. But what are Mersenne primes really and why are they important in finding the largest prime numbers?

If a prime number can be expressed in the form 2^n - 1, n an integer, then it is said to be a Mersenne prime. The name came from Marin Mersenne who studied them, a French monk and well-known mathematician in the 17th century.

mersenneprimes

Many early mathematicians saw and conjectured that if n is a prime number, 2^n - 1 is prime.  For instance, in the figure above, for n = 2, 3, 5, 7 which are prime numbers, their corresponding 2^n - 1 values are also prime numbers. However, this is not true in general since 2^{11} - 1 = 2047 = (23)(89) is not prime.  » Read more

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