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# M9

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, $\displaystyle{ 2^{127}-1 }$ 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, $\displaystyle{ 2^{(2^{61}-1)}-1 }$