I wondered what happened when you take a fraction, a/b, and calculate a/b – (a/b)^2 = c/d. And an interesting pattern appeared when I tried a prime denominator and all numerators less than that denominator. Here’s the pattern with the prime denominator of 7:

1/7 - 01/49 = 06/49
2/7 - 04/49 = 10/49
3/7 - 09/49 = 12/49
4/7 - 16/49 = 12/49
5/7 - 25/49 = 10/49
6/7 - 36/49 = 06/49

And here it is with the prime denominator of 13:

01/13 - 001/169 = 12/169
02/13 - 004/169 = 22/169
03/13 - 009/169 = 30/169
04/13 - 016/169 = 36/169
05/13 - 025/169 = 40/169
06/13 - 036/169 = 42/169
07/13 - 049/169 = 42/169
08/13 - 064/169 = 40/169
09/13 - 081/169 = 36/169
10/13 - 100/169 = 30/169
11/13 - 121/169 = 22/169
12/13 - 144/169 = 12/169

It’s easier to see what’s going on with the smaller denominator:

1/7 - 01/49 = 06/49 = 1/7 - (1/7)^2
2/7 - 04/49 = 10/49 = 2/7 - (2/7)^2
3/7 - 09/49 = 12/49 = 3/7 - (3/7)^2
4/7 - 16/49 = 12/49
5/7 - 25/49 = 10/49
6/7 - 36/49 = 06/49

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1/7 - 01/49 = 07/49 - 01/49 = 06/49
2/7 - 04/49 = 14/49 - 04/49 = 10/49
3/7 - 09/49 = 21/49 - 09/49 = 12/49
4/7 - 16/49 = 28/49 - 16/49 = 12/49
5/7 - 25/49 = 35/49 - 25/49 = 10/49
6/7 - 36/49 = 42/49 - 36/49 = 06/49

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1/7 - 01/49 = 1*7/7^2 - 1^2/7^2 = 07/49 - 01/49 = 06/49
2/7 - 04/49 = 2*7/7^2 - 2^2/7^2 = 14/49 - 04/49 = 10/49
3/7 - 09/49 = 3*7/7^2 - 3^2/7^2 = 21/49 - 09/49 = 12/49
4/7 - 16/49 = 4*7/7^2 - 4^2/7^2 = 28/49 - 16/49 = 12/49
5/7 - 25/49 = 5*7/7^2 - 5^2/7^2 = 35/49 - 25/49 = 10/49
6/7 - 36/49 = 6*7/7^2 - 6^2/7^2 = 42/49 - 36/49 = 06/49

Here’s a set of the patterns using prime denominators from 3 to 13:

1/3 - 1/9 = 2/9
2/3 - 4/9 = 2/9

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1/5 - 01/25 = 4/25
2/5 - 04/25 = 6/25
3/5 - 09/25 = 6/25
4/5 - 16/25 = 4/25

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1/7 - 01/49 = 06/49
2/7 - 04/49 = 10/49
3/7 - 09/49 = 12/49
4/7 - 16/49 = 12/49
5/7 - 25/49 = 10/49
6/7 - 36/49 = 06/49

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01/11 - 001/121 = 10/121
02/11 - 004/121 = 18/121
03/11 - 009/121 = 24/121
04/11 - 016/121 = 28/121
05/11 - 025/121 = 30/121
06/11 - 036/121 = 30/121
07/11 - 049/121 = 28/121
08/11 - 064/121 = 24/121
09/11 - 081/121 = 18/121
10/11 - 100/121 = 10/121

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01/13 - 001/169 = 12/169
02/13 - 004/169 = 22/169
03/13 - 009/169 = 30/169
04/13 - 016/169 = 36/169
05/13 - 025/169 = 40/169
06/13 - 036/169 = 42/169
07/13 - 049/169 = 42/169
08/13 - 064/169 = 40/169
09/13 - 081/169 = 36/169
10/13 - 100/169 = 30/169
11/13 - 121/169 = 22/169
12/13 - 144/169 = 12/169

Then I tried a/b – (a/b)^3. There were no obvious strong patterns, but something caught my eye in the last subtraction for b = 19:

01/19 - 0001/6859 = 0360/6859 = 1/19 - (1/19)^3
02/19 - 0008/6859 = 0714/6859 = 2/19 - (2/19)^3
03/19 - 0027/6859 = 1056/6859 = 3/19 - (3/19)^3
04/19 - 0064/6859 = 1380/6859
05/19 - 0125/6859 = 1680/6859
06/19 - 0216/6859 = 1950/6859
07/19 - 0343/6859 = 2184/6859
08/19 - 0512/6859 = 2376/6859
09/19 - 0729/6859 = 2520/6859
10/19 - 1000/6859 = 2610/6859
11/19 - 1331/6859 = 2640/6859
12/19 - 1728/6859 = 2604/6859
13/19 - 2197/6859 = 2496/6859
14/19 - 2744/6859 = 2310/6859
15/19 - 3375/6859 = 2040/6859
16/19 - 4096/6859 = 1680/6859
17/19 - 4913/6859 = 1224/6859
18/19 - 5832/6859 = 0666/6859

Look at the final subtraction: 18/19 – 5832/6859 = 666/6859. So the Number of the Beast is the numerator when 18/19 is decreased by (18/19)^3. Dropping the need for powers of a/b, I looked for more beastly fraction subtractions using pairs of simplified fractions. Here are a few:

26/35 - 04/31 = 666/1085 = 0.613824885...
29/35 - 05/29 = 666/1015 = 0.656157635...
32/35 - 02/23 = 666/0805 = 0.827329193...
40/41 - 14/31 = 666/1271 = 0.523996853...
35/43 - 13/35 = 666/1505 = 0.442524917...
23/47 - 01/31 = 666/1457 = 0.457103638...
30/47 - 12/41 = 666/1927 = 0.345614946...
31/47 - 01/23 = 666/1081 = 0.616096207...
31/47 - 02/46 = 666/1081 = 0.616096207...
40/47 - 02/19 = 666/0893 = 0.745800672...
[...]

This fraction subtraction is beastly in two ways: 95/103 – 83/97 = 666/9991 = 0.066659994…

I also noticed that 666 can be the numerator in two ways when the denominator is 19 and its powers:

18/19 - 5832/6859 = 666/6859 = 18/19 - (18/19)^3
18/19 + 0324/0361 = 666/0361 = 18/19 + (18/19)^2

So 666 is both the Number of the Decreased and the Number of the Increased. That double pattern is general when you either decrease (b-1)/b by ((b-1)/b)^3 or increase (b-1)/b by ((b-1)/b)^2:

2/3 - 8/27 = 10/27 = 2/3 - (2/3)^3
2/3 + 04/9 = 10/09 = 2/3 + (2/3)^2

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4/5 - 64/125 = 36/125 = 4/5 - (4/5)^3
4/5 + 16/025 = 36/025 = 4/5 + (4/5)^2

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6/7 - 216/343 = 78/343 = 6/7 - (6/7)^2
6/7 + 036/049 = 78/049 = 6/7 + (6/7)^2

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10/11 - 1000/1331 = 210/1331
10/11 + 0100/0121 = 210/0121

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12/13 - 1728/2197 = 300/2197
12/13 + 0144/0169 = 300/0169

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16/17 - 4096/4913 = 528/4913
16/17 + 0256/0289 = 528/0289

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18/19 - 5832/6859 = 666/6859
18/19 + 0324/0361 = 666/0361

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22/23 - 10648/12167 = 990/12167
22/23 + 00484/00529 = 990/00529