Primes In Binary
Created | Updated Jan 30, 2003
The purpose of this is to see whether any patterns emerge from the analysis of the prime numbers 1 through 97 in binary. Several manipulations will be used, mostly chosen arbitrarily.
Manipulation The First - Translate everything into binary, of course.
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Prime/Binary
1/ 0000001
3/ 0000011
5/ 0000101
7/ 0000111
11/0001011
13/0001101
17/0010001
19/0010011
23/0010111
29/0011101
31/0011111
37/0100101
41/0101001
43/0101011
47/0101111
53/0110101
59/0111011
61/0111101
67/1000011
71/1000111
73/1001001
79/1001111
83/1010011
89/1011001
97/1100001
At this point, the only pattern immediately visible is that all prime numbers up to 97 begin and end with the digit 1.
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Manipulation The Second - Line up the binaries, and depending on the relation of each digit to the one below it, insert a new binary (hereafter "newnary") between each pair based on this (keep a note of the base ten version of each newnary). For example, if two numbers were 1010 and 1101, the newnary between them would be 0111. Continue this in steps until there is only one number left. Newnaries are denoted as <newnary> (base 10).
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Newnaries - Step 1
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0000001
<0000010> (2)
0000011
<0000110> (6)
0000101
<0000010> (2)
0000111
<0001100> (12)
0001011
<0000110> (6)
0001101
<0011100> (28)
0010001
<0000010> (2)
0010011
<0000100> (4)
0010111
<0001010> (10)
0011101
<0000010> (2)
0011111
<0111010> (58)
0100101
<0001100> (12)
0101001
<0000010> (2)
0101011
<0000100> (4)
0101111
<0011010> (26)
0110101
<0001110> (14)
0111011
<0000110> (6)
0111101
<1111110> (62)
1000011
<0000100> (4)
1000111
<0001110> (14)
1001001
<0000110> (6)
1001111
<0011100> (28)
1010011
<0001010> (10)
1011001
<0111000> (56)
1100001
Continuing this process for 10 steps produces no immediately visible patterns.
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Manipulation The First - Translate everything into binary, of course.
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Prime/Binary
1/ 0000001
3/ 0000011
5/ 0000101
7/ 0000111
11/0001011
13/0001101
17/0010001
19/0010011
23/0010111
29/0011101
31/0011111
37/0100101
41/0101001
43/0101011
47/0101111
53/0110101
59/0111011
61/0111101
67/1000011
71/1000111
73/1001001
79/1001111
83/1010011
89/1011001
97/1100001
At this point, the only pattern immediately visible is that all prime numbers up to 97 begin and end with the digit 1.
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Manipulation The Second - Line up the binaries, and depending on the relation of each digit to the one below it, insert a new binary (hereafter "newnary") between each pair based on this (keep a note of the base ten version of each newnary). For example, if two numbers were 1010 and 1101, the newnary between them would be 0111. Continue this in steps until there is only one number left. Newnaries are denoted as <newnary> (base 10).
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Newnaries - Step 1
------------------
0000001
<0000010> (2)
0000011
<0000110> (6)
0000101
<0000010> (2)
0000111
<0001100> (12)
0001011
<0000110> (6)
0001101
<0011100> (28)
0010001
<0000010> (2)
0010011
<0000100> (4)
0010111
<0001010> (10)
0011101
<0000010> (2)
0011111
<0111010> (58)
0100101
<0001100> (12)
0101001
<0000010> (2)
0101011
<0000100> (4)
0101111
<0011010> (26)
0110101
<0001110> (14)
0111011
<0000110> (6)
0111101
<1111110> (62)
1000011
<0000100> (4)
1000111
<0001110> (14)
1001001
<0000110> (6)
1001111
<0011100> (28)
1010011
<0001010> (10)
1011001
<0111000> (56)
1100001
Continuing this process for 10 steps produces no immediately visible patterns.
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