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Difference between revisions of "M9"
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| − | The ninth [[Mersenne prime]], <math>2^{61}-1</math> or | + | The ninth [[Mersenne prime]], <math>2^{61}-1</math> or 2 305 843 009 213 693 951. |
| − | 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, ([[Edouard 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 | + | 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, ([[Edouard 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: | ||
| − | *[[Marin Mersenne]] did not have this number on his list of his conjectured [[ | + | *[[Marin Mersenne]] did not have this number on his list of his conjectured [[prime]]s. |
*Lucas was following the conjectured '''[[Double Mersenne number]]''' or slighty narrower '''Catalan-Mersenne number''' sequence. | *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) | *Lucas had started his testing of M12 much earlier than Pervushin, (Lucas started in 1857, at age 15) | ||
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==External links== | ==External links== | ||
*[http://primes.utm.edu/curios/page.php/2305843009213693951.html Prime curios: 2305843009213693951] | *[http://primes.utm.edu/curios/page.php/2305843009213693951.html Prime curios: 2305843009213693951] | ||
| − | [[Category:Mersenne | + | [[Category:Mersenne prime|M09]] |
Revision as of 22:41, 5 February 2019
The ninth Mersenne prime, [math]\displaystyle{ 2^{61}-1 }[/math] or 2 305 843 009 213 693 951.
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, (Edouard 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]
