Multiply Perfect Number of Order 6
Theorem
The number defined as:
- $n = 2^{36} \times 3^8 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
- $\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
- $\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
is multiply perfect of order $6$.
Proof
From Divisor Sum Function is Multiplicative, we may take each prime factor separately and form $\map {\sigma_1} n$ as the product of the divisor sum of each.
Each of the prime factors which occur with multiplicity $1$ will be treated first.
A prime factor $p$ contributes towards the combined $\sigma_1$ a factor $p + 1$.
Hence we have:
\(\ds \map {\sigma_1} {11}\) | \(=\) | \(\ds 12\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3\) |
\(\ds \map {\sigma_1} {19}\) | \(=\) | \(\ds 20\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 5\) |
\(\ds \map {\sigma_1} {43}\) | \(=\) | \(\ds 44\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 11\) |
\(\ds \map {\sigma_1} {61}\) | \(=\) | \(\ds 62\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 31\) |
\(\ds \map {\sigma_1} {83}\) | \(=\) | \(\ds 84\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3 \times 7\) |
\(\ds \map {\sigma_1} {223}\) | \(=\) | \(\ds 224\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^5 \times 7\) |
\(\ds \map {\sigma_1} {331}\) | \(=\) | \(\ds 332\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 83\) |
\(\ds \map {\sigma_1} {379}\) | \(=\) | \(\ds 380\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 5 \times 19\) |
\(\ds \map {\sigma_1} {601}\) | \(=\) | \(\ds 602\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 7 \times 43\) |
\(\ds \map {\sigma_1} {757}\) | \(=\) | \(\ds 758\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 379\) |
\(\ds \map {\sigma_1} {1201}\) | \(=\) | \(\ds 1202\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 601\) |
\(\ds \map {\sigma_1} {7019}\) | \(=\) | \(\ds 7020\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3^3 \times 5 \times 13\) |
\(\ds \map {\sigma_1} {112 \, 303}\) | \(=\) | \(\ds 112 \, 304\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 7019\) |
\(\ds \map {\sigma_1} {898 \, 423}\) | \(=\) | \(\ds 898 \, 424\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 112 \, 303\) |
\(\ds \map {\sigma_1} {616 \, 318 \, 177}\) | \(=\) | \(\ds 616 \, 318 \, 178\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 7^3 \times 898 \, 423\) |
The remaining factors are treated using Divisor Sum of Power of Prime:
- $\map {\sigma_1} {p^k} = \dfrac {p^{k + 1} - 1} {p - 1}$
Thus:
\(\ds \map {\sigma_1} {2^{36} }\) | \(=\) | \(\ds 2 \times 2^{36} - 1\) | Divisor Sum of Power of 2 | |||||||||||
\(\ds \) | \(=\) | \(\ds 137 \, 438 \, 953 \, 471\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 223 \times 616 \, 318 \, 177\) |
\(\ds \map {\sigma_1} {3^8}\) | \(=\) | \(\ds \dfrac {3^9 - 1} {3 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {19 \, 683 - 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9841\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \times 757\) |
\(\ds \map {\sigma_1} {5^5}\) | \(=\) | \(\ds \dfrac {5^6 - 1} {5 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {15 \, 625 - 1} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3906\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3^2 \times 7 \times 31\) |
\(\ds \map {\sigma_1} {7^7}\) | \(=\) | \(\ds \dfrac {7^8 - 1} {7 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {5 \, 764 \, 801 - 1} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 960 \, 800\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^5 \times 5^2 \times 1201\) |
\(\ds \map {\sigma_1} {13^2}\) | \(=\) | \(\ds \dfrac {13^3 - 1} {13 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2197 - 1} {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 183\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 61\) |
\(\ds \map {\sigma_1} {31^2}\) | \(=\) | \(\ds \dfrac {31^3 - 1} {31 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {29 \, 791 - 1} {30}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 993\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 331\) |
Gathering up the prime factors, we have:
- $\map {\sigma_1} n = 2^{37} \times 3^9 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
- $\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
- $\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
By inspection of the multiplicities of the prime factors of $n$ and $\map {\sigma_1} n$, it can be seen that they match for all except for $2$ and $3$.
It follows that $\map {\sigma_1} n = 2 \times 3 \times n = 6 n$.
Hence the result.
$\blacksquare$
Historical Note
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): $100,895,598,169$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $100,895,598,169$