| Currently there may be errors shown on top of a page, because of a missing Wiki update (PHP version and extension DPL3). |
| Topics | Help • Register • News • History • How to • Sequences statistics • Template prototypes |
Difference between revisions of "M12"
(link corr.) |
m |
||
| Line 10: | Line 10: | ||
| foundwith=[[Lucas sequence]]s | | foundwith=[[Lucas sequence]]s | ||
}} | }} | ||
| − | '''M12''', the 12th known [[Mersenne prime]] 2<sup>127</sup>-1, often referred to as M127. This was first proven to be [[prime]] by [[Édouard 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 [http://news.bbc.co.uk/2/hi/technology/7458479.stm 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 [[Raphael Robinson|Robinson]] at [[University of California, Los Angeles]] found [[M13]], 2<sup>521</sup>-1. | + | '''M12''', the 12th known [[Mersenne prime]] 2<sup>127</sup>-1, often referred to as M127. This was first proven to be [[prime]] by [[Édouard 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 [http://news.bbc.co.uk/2/hi/technology/7458479.stm 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 [[Raphael M. Robinson|Robinson]] at [[University of California, Los Angeles]] found [[M13]], 2<sup>521</sup>-1. |
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 (2<sup>1</sup>-1=2), when placed back in the formula (2<sup>2</sup>-1=7) also produces a prime. When this value is tested (2<sup>7</sup>-1=127), another prime is produced. So, Lucas was testing to see if this trend held. | 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 (2<sup>1</sup>-1=2), when placed back in the formula (2<sup>2</sup>-1=7) also produces a prime. When this value is tested (2<sup>7</sup>-1=127), another prime is produced. So, Lucas was testing to see if this trend held. | ||
Latest revision as of 14:52, 19 September 2021
| M12 | |
|---|---|
| Prime class : | |
| Type : | Mersenne prime |
| Formula : | Mn = 2n - 1 |
| Prime data : | |
| Rank : | 12 |
| n-value : | 127 |
| Number : | 170141183460...715884105727 |
| Digits : | 39 |
| Perfect number : | 2126 • (2127-1) |
| Digits : | 77 |
| Discovery data : | |
| Date of Discovery : | 1876-01-10 |
| Discoverer : | Édouard Lucas |
| Found with : | Lucas sequences |
M12, the 12th known Mersenne prime 2127-1, often referred to as M127. This was first proven to be prime by Édouard 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 University of California, Los Angeles found M13, 2521-1.
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 (21-1=2), when placed back in the formula (22-1=7) also produces a prime. When this value is tested (27-1=127), 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 22127-1-1 = 2170 141 183 460 469 231 731 687 303 715 884 105 727-1, 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 3,000,000,000,000 [1], a bit level over 169.4. The current version of Prime95 cannot handle numbers this large, nor can mfaktc.
