Mersenne Prime - History

History

Mersenne primes were considered already by Euclid, who found a connection with the perfect numbers. They are named after 17th-century French scholar Marin Mersenne, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. His list was largely incorrect, as Mersenne mistakenly included M₆₇ and M₂₅₇ (which are composite), and omitted M₆₁, M₈₉, and M₁₀₇ (which are prime). Mersenne gave little indication how he came up with his list.

Édouard Lucas proved in 1876 that M₁₂₇ is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years, and the largest ever calculated by hand. M₆₁ was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that M₆₇ is actually composite. No factor was found until a famous talk by Cole in 1903. Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one. On the other side of the board, he multiplied 193,707,721 × 761,838,257,287 and got the same number, then returned to his seat (to applause) without speaking. A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list.