# Composite Mersenne Number/Examples/M67

## Example of Composite Mersenne Number

$M_{67}$ (that is, $2^{67} - 1$) is a composite number:

 $\ds 2^{67} - 1$ $=$ $\ds 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 927$ $\ds$ $=$ $\ds 193 \, 707 \, 721 \times 761 \, 838 \, 257 \, 287$ $\ds$ $=$ $\ds \left({2 \times 1 \, 445 \, 580 \times 67 + 1}\right) \times \left({2 \times 5 \, 685 \, 360 \, 129 \times 67 + 1}\right)$

## Historical Note

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.