{"id":23541,"date":"2013-07-18T08:30:32","date_gmt":"2013-07-18T12:30:32","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=23541"},"modified":"2014-06-27T12:12:52","modified_gmt":"2014-06-27T17:12:52","slug":"infinitude-mersenne-primes","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/07\/18\/infinitude-mersenne-primes\/","title":{"rendered":"The Infinitude of Mersenne Primes"},"content":{"rendered":"

\"images\"<\/a>A mystery in mathematics is the driving force of a mathematician\u2019s ambition.\u00a0 The thought of discovering something unique and far-reaching brings excitement that demands a mix of creative thought and raw logic.\u00a0\u00a0 Throughout history, mathematicians with various backgrounds have exploited these skills to extend and transcend former thought. \u00a0In the 16th<\/sup> century, one of these mathematicians was a French Minim friar named Marin Mersenne.<\/p>\n

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As stated in previous posts, prime numbers are still very much mysterious.\u00a0 Given a sufficiently large prime number it is difficult to know where the next one will occur in sequence.\u00a0 Mersenne attempted to derive a formula that would represent all prime numbers and give closure to their mystery.\u00a0 Consequently, in trying to discover a unicorn, he instead discovered a set of numbers that now bare his name.<\/p>\n

Mersenne numbers are numbers of the form \"2^p-1\" where p is a prime number.\u00a0 If \"2^p-1\" is also a prime, then it is known as a Mersenne Prime.\u00a0 To date, the largest known prime number anywhere is the 48th<\/sup> Mersenne Prime at 12,978,189 digits in length.\u00a0\u00a0\u00a0 In general, primes become more \u201crare\u201d as numbers become larger.\u00a0 This is also\u00a0true for Mersenne Primes .<\/p>\n

Therefore one can pose the question: \u00a0Is there infinitely many Mersenne Primes?\u00a0\u00a0\u00a0 Like many questions in the field of Number Theory, this is easy to ask, but very hard to prove.\u00a0 It is well known from Euclid\u2019s proof that there exist infinitely many regular primes.\u00a0 However, for some special sets of prime numbers (Mersenne Primes, Sophie-Germain Primes,\u00a0Twin Primes, etc.) this question has never been answered.<\/p>\n

On the website Great Internet Mersenne Prime Search<\/i> (GIMPS<\/i> found here: http:\/\/ow.ly\/n4dgW<\/a>) all of the modern largest Mersenne Primes have been discovered through crowd sourced computation.\u00a0\u00a0\u00a0 Anyone can participate with just a computer connection and willingness to give up a few computational resources.\u00a0\u00a0\u00a0 The 48TH<\/sup> Mersenne Prime is the latest discovery posted on the site, but this can change at any moment.<\/p>\n

In considering the infinitude of Mersenne Primes it would be interesting to see the emergence of an elegant solution.\u00a0\u00a0 It is neat to ponder such possibilities.\u00a0 Could it come from a proof by contradiction?\u00a0 Maybe it will be trivially evident or a consequence of some other discovery. \u00a0\u00a0Can you link it to another conjecture or theorem? \u00a0Whatever the case, problems like these fuel the fire of ambition and spark the imagination of mathematicians, amateur and professional alike.<\/p>\n

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A mystery in mathematics is the driving force of a mathematician\u2019s ambition.\u00a0 The thought of discovering something unique and far-reaching brings excitement that demands a mix of creative thought and raw logic.\u00a0\u00a0 Throughout history, mathematicians with various backgrounds have exploited … Continue reading →<\/span><\/a><\/p>\n

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