Definition:Mersenne Prime/Historical Note

From ProofWiki
Jump to: navigation, search

Historical Note on Mersenne Primes

Mersenne primes are named for Marin Mersenne, who published a book Cogitata Physico-Mathematica in $1644$, in which he claimed that the only primes $p \le 257$ for which $2^p - 1$ is prime are $2, 3, 5, 7, 13, 17, 19, 31, 67, 127$ and $257$.

He was not entirely correct, as shall be seen.


Previous to that, the special nature of these primes had been noted by Euclid, who showed that if $2^n - 1$ is prime, then $2^{n-1} \left({2^n - 1}\right)$ is perfect. The first four primes of this form were known to him.

The fifth one, $M_{13}$, may have been known to Iamblichus in the $4$th century A.D.[1], but this is uncertain, as he does not explicitly demonstrate it. It was definitely known about by 1456.

Pietro Cataldi is supposed to have discovered the $6$th and $7$th Mersenne primes $M_{17}$ and $M_{19}$ in $1588$. Recent researches[2], however, suggest that these may have already been discovered by $1460$. But as no evidence has been found from that date that they had been proven to be prime, it is possible that these were just lucky guesses.

Cataldi also claimed the primality of the Mersenne numbers $M_{23}, M_{29}, M_{31}$ and $M_{37}$.[3] In this he was correct only about $M_{31}$, so that also backs up the suggestion that he was only guessing.


Work started in earnest on these numbers from Mersenne's work.

  • In the $17$th century, Fermat showed that $M_{23}$ has $47$ as a divisor, and that $M_{37}$ has $223$ as a factor.
  • $1738$: Euler showed that $M_{29}$ is composite, having the factor $233$.
  • $1772$: Euler showed that $M_{31}$ is indeed prime.
  • $1903$: The factors of $M_{67}$ were found by Frank Cole who delivered a now famous lecture On The Factorization of Large Numbers in which he performed (without uttering a word) the arithmetic demonstrating what those factors were.


Thus Mersenne's assertion was finally investigated in full: he had been determined to be wrong by:

including $M_{67}$ and $M_{257}$ in his list of primes;
failing to include $M_{61}$, $M_{89}$ and $M_{107}$.

(Pervushin's discovery of the primality of $M_{61}$ caused some to suggest that Mersenne's claim of the primality of $M_{67}$ may have been a copying error for $M_{61}$.)

Nobody will ever know how Mersenne came to his conclusions, as it is impossible with the mathematical knowledge of the time for him to have worked it all out by hand. The fact that he made so few mistakes is incredible.


The work continued, and does so to this day.[4]

  • $1952$: Raphael Robinson used a computer to show that $M_{521}, M_{607}, M_{1279}, M_{2203}$ and $M_{2281}$ are all prime.
  • During the next four decades, the count of known Mersenne primes was doubled by various mathematicians testing supercomputers.

Since then, hunting for Mersenne primes has become a casual hobby for anyone who has access to a computer.


Notable Quotes

We may be able to recognize directly that $5$, or even $17$, is prime, but nobody can convince himself that $2^{127} - 1$ is prime except by studying a proof. No one ever had an imagination so vivid and comprehensive as that.
-- G.H. Hardy


References

  1. 1908: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: footnote to Book $\text{IX}$ Proposition $36$.
  2. Ettore Picutti, in Historia Mathematica pp $123$ - $136$ ($1989$), records that an anonymous author had discovered by $1460$ that both $M_{17}$ and $M_{19}$ are prime.
  3. See The Largest Known Prime by Year.
  4. See Chris Caldwell's website The Prime Pages for more detail of the history of the unearthing of Mersenne primes, and more.


Sources