Years ago when I became interested in ancient measures the first to mark my paper was a gentleman named Garnett Greenbury. He knew I was not a math person but still he was a very unusual gentleman and he said to me if you ever come across this number 5040 please let me know.

It has been staring me in the face for years.

Many people accept 756 imperial feet as a measure for the length of the Great Pyramid and believe it equates to 440 Royal cubits. I don't believe that is correct but in any case I had realised that the diagonal of the Great Pyramid was 3 times that of the base length of the 3rd Pyramid. I. E. S . Edwards the 'Keeper of Egyptian antiquity' at the British museum had written that the base length of the 3rd pyramid = 356.5 feet and this I realised was 1/3rd of the diagonal of the Great Pyramid.

To cut this short and for interest 756 feet produces a diagonal for the Great Pyramid is 1069.145453 imperial feet and 1/3rd of that figure = 356.3818177 Imperial feet.

The diagonal then of the 3rd pyramid = 504 imperial feet and I hope those of you who are interested will enjoy the number. The birth of some remarkable numbers. How did the ancients know this???

https://joedubs.com/5040-the-perfect-number/

Jim

Below: columns are the index n, highly composite numbers, if the number HCN(n) is a superior highly composite number, the index of that number appears in the 3rd column. In the fourth column we have the number

*j*of which HCN(n) is a factorial (

*j*!). In the last column we have the exponents of the prime factors of HCN(n). 5040 = 2^4 × 3² × 5 × 7 = 4.2.1.1. All the preceding HCNs in the list divide 5040 evenly, i.e., they are divisors of 5040. It is a keen number.

Code: Select all

```
n HCN(n) i j MN(HCN(n))
1 1 1 0
2 2 1 2 1
3 4 2
4 6 2 3 11
5 12 3 21
6 24 4 31
7 36 22
8 48 41
9 60 4 211
10 120 5 5 311
11 180 221
12 240 411
13 360 6 321
14 720 6 421
15 840 3111
16 1260 2211
17 1680 4111
18 2520 7 3211
19 5040 8 7 4211
20 7560 3311
21 10080 5211
22 15120 4311
23 20160 6211
24 25200 4221
25 27720 32111
26 45360 4411
27 50400 5221
28 55440 9 42111
29 83160 33111
30 110880 52111
31 166320 43111
32 221760 62111
33 277200 42211
34 332640 53111
35 498960 44111
36 554400 52211
```