Factors of Mersenne Number M67/Historical Note
Historical Note on Factors of Mersenne Number $M_{67}$
While François Édouard Anatole Lucas had demonstrated in $1876$ that $M_{67}$ is composite, he had not established what its divisors are.
The Factors of Mersenne Number $M_{67}$ were demonstrated by Frank Nelson Cole in a famously dramatic presentation On The Factorization of Large Numbers to a meeting of the American Mathematical Society in October $1903$.
When called to give his lecture, he walked to the blackboard, and worked out the calculation, longhand, of $2^{67}$. Then he carefully subtracted $1$.
Moving to another area of the board, he then multiplied out $193, 707, 721 \times 761, 838, 257, 287$.
The numbers matched.
Cole returned to his seat to thunderous applause, having delivered the only lecture in history in which not a word was spoken.
When asked how long it had taken him to find these factors, he reportedly replied:
- Three years of Sundays.
It is noted that Marin Mersenne had originally listed $M_{67}$ as one of the integers of the form $2^p - 1$ to be prime.
Sources
- 1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { I }$ ... (previous) ... (next): Preface
- 1951: Eric Temple Bell: Mathematics: Queen and Servant of Science
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2^{67} - 1$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.12$: Mersenne ($\text {1588}$ – $\text {1648}$)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2^{67} - 1$