Definition:Mersenne Prime/Historical Note

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 $2^{n - 1} \paren {2^n - 1}$ is perfect. The first four primes of this form were known to him.

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

Hudalrichus Regius 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 Regius 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.

Pietro Cataldi 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.

Cataldi 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 Mersenne's work.

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

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

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

• $1811$: Peter Barlow (somewhat short-sightedly, given historical 20-20 hindsight) stated in his book Elementary Investigation of the Theory of Numbers 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.

• $1876$: François Édouard Anatole Lucas proved that $M_{127}$ is prime, and also discovered that $M_{67}$ is actually composite.

• $1883$: Ivan Pervushin proved that $M_{61}$ is 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.

• $1911$: R.E. Powers proved that $M_{89}$ is prime.

• $1914$: R.E. Powers proved that $M_{107}$ is prime, although precedence for this is also claimed by E. Fauquembergue.

• $1916$: R.E. Powers proved that $M_{241}$ is composite.

• $1922$: Maurice Kraitchik proved that $M_{257}$ is actually composite.

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.

• $1952$: 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

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Proposition $36$: Footnote
• 1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { I }$ ... (previous) ... (next): Preface
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $127$
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.12$: Mersenne ($\text {1588}$ – $\text {1648}$)
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $127$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Mersenne numbers
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Mersenne numbers
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $7$: Patterns in Numbers: Euclid