M12

M12, the 12th known Mersenne prime [math]\displaystyle{ 2^{127}-1 }[/math], often referred to as M127. This was first proven to be prime by Edouard Lucas in 1876, after 19 years of hand calculations (although it is claimed that it can be done by hand in 6 months of non-stop work. See video here.) In 1948 it took a computer 25 minutes to run the program to check it. Now it can be done on a "smart phone" in under one second. This was the largest known Mersenne prime until 1952, when Robinson at UCLA found M13, [math]\displaystyle{ 2^{521}-1 }[/math].

Lucas had bypassed M9, M10, and M11. The casual observer might wonder how and why this happened. Lucas was following a sequence (see Double Mersenne number). The first possible Mersenne prime [math]\displaystyle{ (2^{1}-1=2) }[/math], when placed back in the formula [math]\displaystyle{ (2^{2}-1=7) }[/math] also produces a prime. When this value is tested [math]\displaystyle{ (2^{7}-1=127) }[/math], another prime is produced. So, Lucas was testing to see if this trend held.

There have been efforts to again test this trend. With a number so large [math]\displaystyle{ 2^{(2^{127}-1)}-1 = 2^{170141183460469231731687303715884105727}-1 }[/math], there is no hope of running a primality test on this number anytime in the foreseeable future. (It would take about 4,700,000,000,000,000,000,000,000,000,000,000 GHz-days to run the Lucas-Lehmer test on that number. Having the top 250 supercomputers working together at full power on this, it would take 161,600,000 times as long as the universe has existed.) This may be a case of the "Strong law of small numbers". The only way to be certain, is to find a factor, if one exists. Landon Curt Noll has trial factored this number up to a k value of at least 3000000000000 [1], a bit level over 169.4. The current version of Prime95 cannot handle numbers this large, nor can mfaktc.