Definition:Mersenne Prime/Historical Note

Historical Note on Mersenne Primes
Mersenne primes are named for, who published a book 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, who showed that if $2^n - 1$ is prime, then $2^{n - 1} \paren {2^n - 1}$ is perfect. The first four primes of this form were known to him.

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

showed in $1536$ that not all numbers of the form $2^n - 1$ for odd $n$ are prime numbers.

Up until this time it was assumed so, but gave the counterexamples $2^9 - 1 = 511 = 7 \times 73$ and $2^{11} - 1 = 2047 = 23 \times 89$.

Hence he also demonstrated that it is not even sufficient for $n$ to be prime for $2^n - 1$ also to be prime.

is supposed to have discovered the $6$th and $7$th Mersenne primes $M_{17}$ and $M_{19}$ in $1588$. Recent researches, 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.

also claimed the primality of the Mersenne numbers $M_{23}, M_{29}, M_{31}$ and $M_{37}$. 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 's work.


 * $1640$: showed that $M_{23}$ has $47$ as a divisor, and that $M_{37}$ has $223$ as a divisor.


 * $1738$: showed that $M_{29}$ is composite, having the divisors $233$ and $1103$.


 * $1772$: showed that $M_{31}$ is indeed prime.


 * $1811$: (somewhat short-sightedly, given historical 20-20 hindsight) stated in his book  that $M_{31}$:
 * is the greatest [i.e. prime number] at present known to be such ... and probably the greatest that ever will be discovered. See Barlow's Prediction.


 * $1876$: proved that $M_{127}$ is prime, and also discovered that $M_{67}$ is actually composite.


 * $1883$: proved that $M_{61}$ is prime.


 * $1903$: The factors of $M_{67}$ were found by 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.


 * $1911$: proved that $M_{89}$ is prime.


 * $1914$: proved that $M_{107}$ is prime.


 * $1916$: proved that $M_{241}$ is composite.


 * $1922$: proved that $M_{257}$ is actually composite.

Thus '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 $M_{107}$.

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

Nobody will ever know how 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.


 * $1952$: 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.