Part II is here
I've watched Dr James Grimes explain Why 82,000 is an extraordinary number.
I was intrigued by this sequence and disocvered the following patterns.
I was wondering if I had just discovered the "Obvious" or stumbled across a possible method to identify the next number or prove there isnt a next number.
What I discovered is as follows:-
Base 3
Within base 3 counterfeit binary numbers (e.g. numbers comprising of digits 1 and 0) greater that 1 always appear in pairs, e.g. 10, 11, then 100, 101 etc.. By counting the base 3 numbers that occur consecutively containing the digit 2 we see a pattern emerge, e.g. 1, 4, 1, 13, 1, 4 etc.
This table shows (i) The first 39 base 3 numbers (0 - 38), (ii) The counterfeit binary numbers always appear in consecutive pairs, (iii) The start of the digit 2 occurrence pattern.
Decimal Counterfeit Digit 2 Count
0 0
1 1
2 2 1
3 10
4 11
5 12
6 20
7 21
8 22 4
9 100
10 101
11 102 1
12 110
13 111
14 112
15 120
16 121
17 122
18 200
19 201
20 202
21 210
22 211
23 212
24 220
25 221
26 222 13
27 1000
28 1001
29 1002 1
30 1010
31 1011
32 1012
33 1020
34 1021
35 1022 4
36 1100
37 1101
38 1102
The base 3 pattern that's associated with the base 3 digit 2 occurrence pattern has a cycle of 64 and is as follows:-
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 3280,
The numbers within each line have the following properties. The lines contain both a horizontal and vertical pattern
Horizontal Pattern
All odd positions within each line are 1. The even position maximum numbers are positioned at powers of 2 from left to right e.g.
Position Maximum
2 4
4 13
8 40
16 121
32 364
64 1093
The maximums are calculated
(3 * n) + 1, 4 = (3 * 1) + 1, 13 = (3 * 4) + 1, 40 = (3 * 13) + 1, 121 = (3 * 40) + 1, 364 = (3 * 121) + 1, 1093=(3 * 364) + 1
Vertical Pattern
The 64th number in each line follows a cyclic pattern.
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 3280,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 9841,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 3280,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 29524,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 3280,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 9841,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 1093,
1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 364, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13, 1, 4, 1, 121, 1, 4, 1, 13, 1, 4, 1, 40, 1, 4, 1, 13
The vertical maximums appears on Line 2, Line 4, Line 8, Line 16 etc. each maximum is calculated
(3 * n) + 1, 3280 = (3 * 1093) + 1, 9841 = (3 * 3280) + 1, 29524 = (3 * 9841) + 1
Base 4
As with Base 3, within base 4 counterfeit binary numbers greater that 1 always appear in pairs, e.g. 10, 11, then 100, 101 etc.. By counting the base 4 numbers that occur consecutively containing digits 2 and/or 3 we see a pattern emerge, e.g. 2, 10, 2, 42, 2, 10 etc.
This table shows (i) The first 39 base 4 numbers (0 - 38), (ii) The counterfeit binary numbers always appear in consecutive pairs, (iii) The start of the digits 2 and 3 occurrence pattern.
Decimal Counterfeit Digit 2 & 3 Count
0 0
1 1
2 2
3 3 2
4 10
5 11
6 12
7 13
8 20
9 21
10 22
11 23
12 30
13 31
14 32
15 33 10
16 100
17 101
18 102
19 103 2
20 110
21 111
22 112
23 113
24 120
25 121
26 122
27 123
28 130
29 131
30 132
31 133
32 200
33 201
34 202
35 203
36 210
37 211
38 212
The base 4 pattern that's associated with the base 4 digits 2 and/or 3 occurrence pattern has a cycle of 64 and is as follows:-
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 43690,
The numbers within each line have the following properties. The lines contain both a horizontal and vertical pattern
Horizontal Pattern
All odd positions within each line are 2. The even maximum numbers are positioned at powers of 2 from left to right e.g.
Position Maximum
2 10
4 42
8 170
16 682
32 2730
64 10922
The maximums are calculated
(4 * n) + 2, 10 = (4 * 2) + 2, 42 = (4 * 10) + 2, 170 = (4 * 42) + 2, 682 = (4 * 170) + 2, 2730 = (4 * 682) + 2, 10922 = (4 * 2730) + 2
Vertical Pattern
The 64th number in each line follows a cyclic pattern.
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 43690,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 174762,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 43690,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 699050,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 43690,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 174762,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 43690,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 10922,
2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2730, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 682, 2, 10, 2, 42, 2, 10, 2, 170, 2, 10, 2, 42, 2, 10, 2, 2796202
The maximums appears on Line 2, Line 4, Line 8, Line 16 etc. each maximum is calculated
(4 * n) + 2, 43690 = (4 * 10922) + 2, 174762 = (4 * 43690) + 2, 699050 = (4 * 174762) + 2, 2796202 = (4 * 699050) + 2
Base 5
As with Base 3 and 4, within base 5 counterfeit binary numbers greater that 1 always appear in pairs, e.g. 10, 11, then 100, 101 etc.. By counting the base 5 numbers that occur consecutively containing digits 2, 3 and/or 4 we see a pattern emerge, e.g. 3, 18, 3, 93, 3, 18, 3, 468, 3 etc.
This table shows (i) The first 39 base 5 numbers (0 - 38), (ii) The counterfeit binary numbers always appear in consecutive pairs, (iii) The start of the digit 2,3 and/or 4 occurrence pattern.
Decimal Counterfeit Digit 2,3 & 4 Count
0 0
1 1
2 2
3 3
4 4 3
5 10
6 11
7 12
8 13
9 14
10 20
11 21
12 22
13 23
14 24
15 30
16 31
17 32
18 33
19 34
20 40
21 41
22 42
23 43
24 44 18
25 100
26 101
27 102
28 103
29 104 3
30 110
31 111
32 112
33 113
34 114
35 120
36 121
37 122
38 123
The base 5 pattern that's associated with the base 5 digits 2, 3 and/or 4 occurrence pattern has a cycle of 64 and is as follows:-
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 292968,
The numbers within each line have the following properties. The lines contain both a horizontal and vertical pattern
Horizontal Pattern
All odd positions within each line are 3.
The even position maximum numbers are positioned at powers of 2 from left to right e.g.
Position Maximum
2 18
4 93
8 468
16 2343
32 11718
64 58593
The maximums are calculated
(5 * n) + 3, 18 = (5 * 3) + 3, 93 = (5 * 18) + 3, 468 = (5 * 93) + 3, 2343 = (5 * 468) + 3, 11718 = (5 * 2343) + 3, 58593 = (5 * 11718) + 3
Vertical Pattern
The 64th number in each line follows a cyclic pattern.
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 292968,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 1464843,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 292968,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 7324218,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 292968,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 1464843,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 292968,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 36621093,
3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 11718, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 2343, 3, 18, 3, 93, 3, 18, 3, 468, 3, 18, 3, 93, 3, 18, 3, 58593,
The maximums appears on Line 2, Line 4, Line 8, Line 16 etc. each maximum is calculated
(5 * n) + 3, 292968 = (5 * 58593) + 3, 1464843 = (5 * 292968) + 3, 7324218 = (5 * 1464843) + 3, 36621093 = (5 * 7324218) + 3
Base 6
As with Base 3, 4 and 5, within base 6 counterfeit binary numbers greater that 1 always appear in pairs, e.g. 10, 11, then 100, 101 etc.. By counting the base 6 numbers that occur consecutively containing digits 2, 3, 4 and/or 5 we see a pattern emerge, e.g. 4, 28, 4, 172, 4, 28, 4, 1036, 4 etc.
This table shows (i) The first 39 base 6 numbers (0 - 38), (ii) The counterfeit binary numbers always appear in consecutive pairs, (iii) The start of the digit 2,3, 4 and/or 5 occurrence pattern.
Decimal Counterfeit Digit 2,3,4 & 5 Count
0 0
1 1
2 2
3 3
4 4
5 5 4
6 10
7 11
8 12
9 13
10 14
11 15
12 20
13 21
14 22
15 23
16 24
17 25
18 30
19 31
20 32
21 33
22 34
23 35
24 40
25 41
26 42
27 43
28 44
29 45
30 50
31 51
32 52
33 53
34 54
35 55 28
36 100
37 101
38 102
The base 6 pattern that's associated with the base 6 digits 2, 3, 4 and/or 5 occurrence pattern has a cycle of 32 and is as follows:-
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 223948,
The numbers within each line have the following properties. The lines contain both a horizontal and vertical pattern
Horizontal Pattern
All odd positions within each line are 4.
The even position maximum numbers are positioned at powers of 2 from left to right e.g.
Position Maximum
2 28
4 172
8 1036
16 6220
32 37324
The maximums are calculated
(6 * n) + 4, 28 = (6 * 4) + 4, 172 = (6 * 28) + 4, 1036 = (6 * 172) + 4, 6220 = (6 * 1036) + 4, 37324 = (6 * 6220) + 4
Vertical Pattern
The 32nd number in each line follows a cyclic pattern.
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 223948,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 1343692,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 223948,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 8062156,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 223948,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 1343692,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 223948,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 37324,
4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 6220, 4, 28, 4, 172, 4, 28, 4, 1036, 4, 28, 4, 172, 4, 28, 4, 48372940,
The maximums appears on Line 2, Line 4, Line 8, Line 16 etc. each maximum is calculated
(6 * n) + 4, 223948 = (6 * 37324) + 4, 1343692 = (6 * 223948) + 4, 8062156 = (6 * 1343692) + 4, 48372940 = (6 * 8062156) + 4