Multiply Perfect Number of Order 8
Theorem
The number defined as:
- $n = 2^{65} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
- $\times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$
- $\times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$
- $\times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$
is multiply perfect of order $8$.
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:
\(\ds \map {\sigma_1} {23}\) | \(=\) | \(\ds 24\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3\) |
\(\ds \map {\sigma_1} {37}\) | \(=\) | \(\ds 38\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 19\) |
\(\ds \map {\sigma_1} {41}\) | \(=\) | \(\ds 42\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 7\) |
\(\ds \map {\sigma_1} {53}\) | \(=\) | \(\ds 54\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3^3\) |
\(\ds \map {\sigma_1} {61}\) | \(=\) | \(\ds 62\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 31\) |
\(\ds \map {\sigma_1} {73}\) | \(=\) | \(\ds 74\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 37\) |
\(\ds \map {\sigma_1} {83}\) | \(=\) | \(\ds 84\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3 \times 7\) |
\(\ds \map {\sigma_1} {89}\) | \(=\) | \(\ds 90\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3^2 \times 5\) |
\(\ds \map {\sigma_1} {103}\) | \(=\) | \(\ds 104\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 13\) |
\(\ds \map {\sigma_1} {127}\) | \(=\) | \(\ds 128\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^7\) |
\(\ds \map {\sigma_1} {131}\) | \(=\) | \(\ds 132\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3 \times 11\) |
\(\ds \map {\sigma_1} {149}\) | \(=\) | \(\ds 150\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 5^2\) |
\(\ds \map {\sigma_1} {211}\) | \(=\) | \(\ds 212\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 53\) |
\(\ds \map {\sigma_1} {307}\) | \(=\) | \(\ds 308\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 7 \times 11\) |
\(\ds \map {\sigma_1} {331}\) | \(=\) | \(\ds 332\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 83\) |
\(\ds \map {\sigma_1} {463}\) | \(=\) | \(\ds 464\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 29\) |
\(\ds \map {\sigma_1} {521}\) | \(=\) | \(\ds 522\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3^2 \times 29\) |
\(\ds \map {\sigma_1} {683}\) | \(=\) | \(\ds 684\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3^2 \times 19\) |
\(\ds \map {\sigma_1} {709}\) | \(=\) | \(\ds 710\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 5 \times 71\) |
\(\ds \map {\sigma_1} {1279}\) | \(=\) | \(\ds 1280\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^8 \times 5\) |
\(\ds \map {\sigma_1} {2141}\) | \(=\) | \(\ds 2142\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3^2 \times 7 \times 17\) |
\(\ds \map {\sigma_1} {2557}\) | \(=\) | \(\ds 2558\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 1279\) |
\(\ds \map {\sigma_1} {5113}\) | \(=\) | \(\ds 5114\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 2557\) |
\(\ds \map {\sigma_1} {6481}\) | \(=\) | \(\ds 6482\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 7 \times 463\) |
\(\ds \map {\sigma_1} {10 \, 429}\) | \(=\) | \(\ds 10 \, 430\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 5 \times 7 \times 149\) |
\(\ds \map {\sigma_1} {20 \, 857}\) | \(=\) | \(\ds 20 \, 858\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 10 \, 429\) |
\(\ds \map {\sigma_1} {110 \, 563}\) | \(=\) | \(\ds 110 \, 564\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 131 \times 211\) |
\(\ds \map {\sigma_1} {599 \, 479}\) | \(=\) | \(\ds 599 \, 480\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 5 \times 7 \times 2141\) |
\(\ds \map {\sigma_1} {16 \, 148 \, 168 \, 401}\) | \(=\) | \(\ds 16 \, 148 \, 168 \, 402\) | Divisor Sum of Prime Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 103 \times 709 \times 110 \, 563\) |
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:
\(\ds \map {\sigma_1} {2^{65} }\) | \(=\) | \(\ds 2 \times 2^{65} - 1\) | Divisor Sum of Power of 2 | |||||||||||
\(\ds \) | \(=\) | \(\ds 73 \, 786 \, 976 \, 294 \, 838 \, 206 \, 463\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3^2 \times 7 \times 23 \times 67 \times 89 \times 683 \times 20 \, 857 \times 599 \, 479\) |
\(\ds \map {\sigma_1} {3^{23} }\) | \(=\) | \(\ds \dfrac {3^{24} - 1} {3 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {282 \, 429 \, 536 \, 481 - 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 141 \, 214 \, 768 \, 240\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 5 \times 7 \times 13 \times 41 \times 73 \times 6481\) |
\(\ds \map {\sigma_1} {5^9}\) | \(=\) | \(\ds \dfrac {5^{10} - 1} {5 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {9 \, 765 \, 625 - 1} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \, 441 \, 406\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 11 \times 71 \times 521\) |
\(\ds \map {\sigma_1} {7^{12} }\) | \(=\) | \(\ds \dfrac {7^{13} - 1} {7 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {96 \, 889 \, 010 \, 407 - 1} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 16 \, 148 \, 168 \, 401\) |
\(\ds \map {\sigma_1} {11^3}\) | \(=\) | \(\ds \dfrac {11^4 - 1} {11 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {14 \, 641 - 1} {10}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1464\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3 \times 61\) |
\(\ds \map {\sigma_1} {13^3}\) | \(=\) | \(\ds \dfrac {13^4 - 1} {13 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {28 \, 561 - 1} {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2380\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 5 \times 7 \times 17\) |
\(\ds \map {\sigma_1} {17^2}\) | \(=\) | \(\ds \dfrac {17^3 - 1} {17 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4913 - 1} {16}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 307\) |
\(\ds \map {\sigma_1} {19^2}\) | \(=\) | \(\ds \dfrac {19^3 - 1} {19 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {6859 - 1} {18}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 381\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 127\) |
\(\ds \map {\sigma_1} {29^2}\) | \(=\) | \(\ds \dfrac {29^3 - 1} {29 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {24 \, 389 - 1} {28}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 871\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \times 67\) |
\(\ds \map {\sigma_1} {31^2}\) | \(=\) | \(\ds \dfrac {31^3 - 1} {31 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {29 \, 791 - 1} {30}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 993\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 331\) |
\(\ds \map {\sigma_1} {67^2}\) | \(=\) | \(\ds \dfrac {67^3 - 1} {67 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {300 \, 763 - 1} {66}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4557\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 7^2 \times 31\) |
\(\ds \map {\sigma_1} {71^2}\) | \(=\) | \(\ds \dfrac {71^3 - 1} {71 - 1}\) | Divisor Sum of Power of Prime | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {357 \, 911 - 1} {66}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5113\) |
Gathering up the prime factors, we have:
- $\map {\sigma_1} n = 2^{68} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
- $\times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$
- $\times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$
- $\times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$
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 $2$.
It follows that $\map {\sigma_1} n = 2^3 \times n = 8 n$.
Hence the result.
$\blacksquare$
Sources
- 1954: Alan L. Brown: Multiperfect Numbers (Scripta Math Vol. 20: pp. 103 – 106)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $120$
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $120$