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Difference between revisions of "M9"
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| discovery=1883 | | discovery=1883 | ||
| discoverer=[[Ivan Mikheevich Pervushin]] | | discoverer=[[Ivan Mikheevich Pervushin]] | ||
| + | | foundwith=[[Lucas sequence]]s | ||
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The ninth [[Mersenne prime]], 2<sup>61</sup>-1 or {{Num|2305843009213693951}}. | The ninth [[Mersenne prime]], 2<sup>61</sup>-1 or {{Num|2305843009213693951}}. | ||
| − | It was determined to be prime in 1883 by [http://en.wikipedia.org/wiki/Ivan_Mikheevich_Pervushin Ivan Mikheevich Pervushin] and for this reason it is sometimes called '''Pervushin's number'''. At the time of Pervushin's proof it was the second-largest known prime number, ([[ | + | It was determined to be prime in 1883 by [http://en.wikipedia.org/wiki/Ivan_Mikheevich_Pervushin Ivan Mikheevich Pervushin] and for this reason it is sometimes called '''Pervushin's number'''. At the time of Pervushin's proof it was the second-largest known prime number, ([[Édouard Lucas]] having shown earlier that [[M12]], <math>2^{127}-1</math> is also prime), and it remained so until 1911. Prior to the developement of the Lucas test all Mersenne primes were proved by some form of [[trial factoring]]. Pervushin used the [[Lucas-Lehmer test]] to prove that this number is prime. |
The reasons that lead to it's discovery out of order: | The reasons that lead to it's discovery out of order: | ||
Latest revision as of 14:30, 17 February 2019
| M9 | |
|---|---|
| Prime class : | |
| Type : | Mersenne prime |
| Formula : | Mn = 2n - 1 |
| Prime data : | |
| Rank : | 9 |
| n-value : | 61 |
| Number : | 2305843009213693951 |
| Digits : | 19 |
| Perfect number : | 260 • (261-1) |
| Digits : | 37 |
| Discovery data : | |
| Date of Discovery : | 1883 |
| Discoverer : | Ivan Mikheevich Pervushin |
| Found with : | Lucas sequences |
The ninth Mersenne prime, 261-1 or 2,305,843,009,213,693,952.
It was determined to be prime in 1883 by Ivan Mikheevich Pervushin and for this reason it is sometimes called Pervushin's number. At the time of Pervushin's proof it was the second-largest known prime number, (Édouard Lucas having shown earlier that M12, [math]\displaystyle{ 2^{127}-1 }[/math] is also prime), and it remained so until 1911. Prior to the developement of the Lucas test all Mersenne primes were proved by some form of trial factoring. Pervushin used the Lucas-Lehmer test to prove that this number is prime.
The reasons that lead to it's discovery out of order:
- Marin Mersenne did not have this number on his list of his conjectured primes.
- Lucas was following the conjectured Double Mersenne number or slighty narrower Catalan-Mersenne number sequence.
- Lucas had started his testing of M12 much earlier than Pervushin, (Lucas started in 1857, at age 15)
Of note is the fact that to date (2011): the smallest Double Mersenne number with an unknown status is MM61, [math]\displaystyle{ 2^{(2^{61}-1)}-1 }[/math]
