The Handy Math Answer Book

85

MATH BASICS

What is the story behind the Mersenne primes?

M

ersenne primes (or Mersenne numbers) are connected to prime numbers.
They come in the form of 2p1, in which pis a prime; or, to put it another
way, when 2p1 is prime, it is said to be a Mersenne prime.

Centuries ago, many mathematicians believed that numbers from the form
2 p1 (they actually used the form 2n1, which is the same as the 2p1 used
today) were prime for all primes p. By the 16th century, it was proven that 2^11 
1 2,047 was not prime. By 1603 Pietro Cataldi (1548–1626) correctly discov-
ered that p17 and p19 were both prime, but he was wrong to add 23, 29,
and 37 to his prime numbers list. Soon, others discovered his errors, including
French mathematician Pierre de Fermat (1601–1665) in 1640 and Swiss mathe-
matician Leonhard Euler (1707–1783) in 1738.

The hunt for primes continued. The name “Mersenne” actually came from the
French priest Father Marin Mersenne (1588–1648), who in 1644 referred to such
numbers in the preface to his book Cogitata Physica-Mathematica. He believed
that these special primes were: p2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257. But
like earlier attempts at determining prime numbers, many of Mersenne’s numbers
were in error. It took three centuries more to check Mersenne’s range of numbers,
and by 1947 the correct list of Mersenne primes were: p2, 3, 5, 7, 13, 17, 19, 31,
61, 89, 107 and 127. Interestingly enough, even though Mersenne incorrectly stat-
ed that certain numbers belonged to this group—he probably didn’t verify all the
numbers on his list—his name is still attached to these numbers.

Mersenne Year Mersenne Year Mersenne Year
Prime Discovered Prime Discovered Prime Discovered

2—

3—

5—

7—

13 1456

17 1588

19 1588

31 1772

61 1883

89 1911

107 1914

127 1876

521 1952

607 1952

1,279 1952

2,203 1952

2,281 1952

3,217 1957

4,253 1961

4,423 1961

9,689 1963

9,941 1963

11,213 1963

19,937 1971

21,701 1978

23,209 1979

44,497 1979

86,243 1982

110,503 1988

132,049 1983

216,091 1985

756,839 1992

859,433 1994

1,257,787 1996

1,398,269 1996

2,976,221 1997

3,021,377 1998

6,972,593 1999

13,466,917* 2001

20,996,011* 2003

24,036,583* 2004

*These numbers have not yet been confirmed as Mersenne primes by the Great Internet

Mersenne Prime Search at press time (GIMPS; see boxed text).