# Multiply Perfect Number of Order 6/Historical Note

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## Historical Note on Multiply Perfect Number of Order 6

Marin Mersenne, in a letter of $1643$, challenged Pierre de Fermat to find the ratio of:

- $2^{36} \times 3^8 \times 5^5 \times 11 \times 13^2 \times 19 \times 31^2$
- $\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
- $\times \ 7019 \times 823 \, 543 \times 616 \, 318 \, 177 \times 100 \, 895 \, 598 \, 169$

to its aliquot sum.

Fermat replied that its ratio to the sum of all its divisors (including the number itself) was $6$.

He also pointed out that $100 \, 895 \, 598 \, 169 = 112 \, 303 \times 898 \, 423$, both of which divisors are prime.

Also note that $823 \, 543 = 7^7$, another point that Marin Mersenne glossed over, intentionally or inadvertently, in his initial challenge.

Both Mersenne's initial challenge and Fermat's factorisation of $100 \, 895 \, 598 \, 169$ were remarkable, considering the lack of computing machines in those days. To this day, nobody knows how they did it.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $100,895,598,169$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $100,895,598,169$