The Infinitude of Mersenne Primes

A mystery in mathematics is the driving force of a mathematician’s ambition.  The thought of discovering something unique and far-reaching brings excitement that demands a mix of creative thought and raw logic.   Throughout history, mathematicians with various backgrounds have exploited these skills to extend and transcend former thought.  In the 16^th century, one of these mathematicians was a French Minim friar named Marin Mersenne.

As stated in previous posts, prime numbers are still very much mysterious.  Given a sufficiently large prime number it is difficult to know where the next one will occur in sequence.  Mersenne attempted to derive a formula that would represent all prime numbers and give closure to their mystery.  Consequently, in trying to discover a unicorn, he instead discovered a set of numbers that now bare his name.

Mersenne numbers are numbers of the form where p is a prime number.  If is also a prime, then it is known as a Mersenne Prime.  To date, the largest known prime number anywhere is the 48^th Mersenne Prime at 12,978,189 digits in length.    In general, primes become more “rare” as numbers become larger.  This is also true for Mersenne Primes .

Therefore one can pose the question:  Is there infinitely many Mersenne Primes?    Like many questions in the field of Number Theory, this is easy to ask, but very hard to prove.  It is well known from Euclid’s proof that there exist infinitely many regular primes.  However, for some special sets of prime numbers (Mersenne Primes, Sophie-Germain Primes, Twin Primes, etc.) this question has never been answered.

On the website Great Internet Mersenne Prime Search (GIMPS found here: http://ow.ly/n4dgW) all of the modern largest Mersenne Primes have been discovered through crowd sourced computation.    Anyone can participate with just a computer connection and willingness to give up a few computational resources.    The 48^TH Mersenne Prime is the latest discovery posted on the site, but this can change at any moment.

In considering the infinitude of Mersenne Primes it would be interesting to see the emergence of an elegant solution.   It is neat to ponder such possibilities.  Could it come from a proof by contradiction?  Maybe it will be trivially evident or a consequence of some other discovery.   Can you link it to another conjecture or theorem?  Whatever the case, problems like these fuel the fire of ambition and spark the imagination of mathematicians, amateur and professional alike.