Programmer Needed

[Sigma 768]

Okay, so I decided to take a break from RSA numbers and just focus on the primes themselves. I thought it would be best if I did some Wikipedia searches to refresh my memory to make sure I wasn't missing the guerilla in the room (this video will show you what I'm talking about). Well, I did some reading, and I read something thought provoking that I have discerned ever since I started my experimenting:

 

"The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 × 3, 1 × 1 × 3, etc. are all valid factorizations of 3."

-http://en.wikipedia.org/wiki/Prime_number

 

Now, don't get me wrong. I've known about this rule since I first started, but I think this rule has a lot more potential than is being played out. Let's say, for instance, we bend the rule and make it to where both 1 and 2 are excluded. What would that do?

 

Well, like 1, 2 will no longer be considered prime nor composite, and we won't be able to use it in prime factorization. You'd think this means that every composite number that has a 2 in it's prime factorization would turn into a prime, right? Well, often times yes, but this isn't always the case. Now, I want you to think of the prime factorization of 4, which is 2x2. Because we bent the rule to where 2 is like 1, we cannot use it in factorizing 4. Therefore, 4 is now a prime number, and can be used to factorize certain composite numbers that required 2 in order to be constant. An example of this would be 2x2x2x2=16, where, when we bent the rule, 2x2x2x2 turned into 4x4, and 16 was allowed to remain a constant.

 

I experimented with this idea all the way up to excluding 1 through 4, and finding the prime factorization of numbers 1 to 99. I'm not claiming that all of my data is accurate considering that my brain is just now getting used to this concept, but there are several instances where you can have two valid factorizations of a composite number. For instance, number 90 when excluding 1, 2, and 3. You can get the factorization of either 6x15 or 5x18, since you cannot use 3 to get 3x5=15, or use both 2 and 3 to get 2x3x3=18 or 2x3=6.

 

This is the part where a fellow brony who can write programs comes in to help me. I would like to not only have my work tested and expanded upon with a computer program, but I would also appreciate it if you made a visual representation of the data as well. Allow me to explain.

 

Since we're only working with natural numbers here, you don't have to make a formal number line. In fact, it would be better if you where to make a line of connected, square blocks with black outlines that each represented a natural number, and have the insides of the squares colour-coated accordingly:

 

White for an excluded number.

Black for a prime number.

Blue for a composite number.

Red for a composite number that has a total of two valid factorizations.

Yellow for a composite number that has a total of three valid factorizations.

 

And so on. Now for the interesting part. Not only do I want you to make a strip of colour-coated blocks that each represent a natural number, but I also want you to make several strips that each represent different scenarios where different numbers are excluded (first just exclude 1; then 1 and 2; then 1, 2, and 3; and so on). Once that's done, you need to place these strips of blocks on top of each other so we can see if any kind of patterns emerge. It would also be very useful if you could provide me with Ulam Spirals that follow the same colour-coated pattern.

 

Why am I not doing this myself you ask? Well, I can't program worth anything, the Ubuntu Linux version of paint tends to crash, the classic 'It would take too long.' excuse, and the fact that doing everything by hand while I'm still getting used to the concept leaves a large margin for error.

 

So, does anyone want to take this responsibility?

 

My work since for some reason I cannot attach any files:

 

Excluding:
 
   1           1 & 2        1, 2, & 3   1, 2, 3, & 4
1  X           X            X           X
2              X            X           X
3                           X           X
4  2x2                                  X
5                                        
6  2x3                                   
7                                        
8  2x2x2                                 
9  3x3         3x3                       
10 2x5                                   
11                                       
12 2x2x3       3x4                       
13                                       
14 2x7                                   
15 3x5         3x5                       
16 2x2x2x2     4x4          4x4          
17                                       
18 2x3x3       3x6                       
19                                       
20 2x2x5       4x5          4x5          
21 3x7         3x7                       
22 2x11                                  
23                                       
24 2x2x2x3     4x6          4x6          
25 5x5         5x5          5x5         5x5
26 2x13                                  
27 3x3x3       3x3x3                     
28 2x2x7       4x7          4x7          
29                                       
30 2x3x5       5x6, 3x10    5x6         5x6
31                                       
32 2x2x2x2x2   4x8          4x8          
33 3x11        3x11                      
34 2x17                                  
35 5x7         5x7          5x7         5x7
36 2x2x3x3     3x3x4, 6x6   6x6, 4x9    6x6
37                                       
38 2x19                                  
39 3x13        3x13                      
40 2x2x2x5     5x8, 4x10    5x8, 4x10   5x8
41                                       
42 2x3x7       6x7, 3x14    6x7         6x7
43                                       
44 2x2x11      4x11         4x11         
45 3x3x5       3x3x5        5x9         5x9
46 2x23                                  
47                                       
48 2x2x2x2x3   3x4x4        4x12         
49 7x7         7x7          7x7         7x7
50 2x5x5       5x10         5x10        5x10
51 3x17        3x17                      
52 2x2x13      4x13         4x13         
53                                       
54 2x3x3x3     6x3x3        6x9         6x9
55 5x11        5x11         5x11        5x11
56 2x2x2x7     4x14, 7x8    4x14, 7x8   7x8
57 3x19        3x19                      
58 2x29                                  
59                                       
60 2x2x3x5     3x4x5, 6x10  6x10        6x10
61                                       
62 2x31                                  
63 3x3x7       3x3x7        7x9         7x9
64 2x2x2x2x2x2 4x4x4, 8x8   4x4x4, 8x8  8x8
65 5x13        5x13         5x13        5x13
66 2x3x11      6x11         6x11        6x11
67                                       
68 2x2x17      4x17         4x17         
69 3x23        3x23                      
70 2x5x7       7x10, 5x14   7x10, 5x14  7x10, 5x14
71                                       
72 2x2x2x3x3   3x4x6, 3x3x8 8x9, 6x12   8x9, 6x12
73                                       
74 2x37                                  
75 3x5x5       3x5x5        5x15        5x15
76 2x2x19      4x19         4x19         
77 7x11        7x11         7x11        7x11
78 2x3x13      6x13         6x13        6x13
79                                       
80 2x2x2x2x5   4x4x5, 8x10  4x4x5, 8x10 8x10
81 3x3x3x3     3x3x3x3      9x9         9x9
82 2x41                                  
83                                       
84 2x2x3x7     3x4x7        7x12, 4x21  7x12
85 5x17        5x17         5x17        5x17
86 2x43                                  
87 3x29        3x29                      
88 2x2x2x11    8x11, 4x22   8x11, 4x22  8x11
89                                       
90 2x3x3x5     3x5x6        6x15, 5x18  6x15, 5x18
91 7x13        7x13         7x13        7x13
92 2x2x23      4x23         4x23         
93 3x31        3x31                      
94 2x47                                  
95 5x19        5x19         5x19        5x19
96 2x2x2x2x2x3 4x4x6, 6x16  4x4x6, 6x16 6x16
97                                       
98 2x7x7       7x14         7x14        7x14
99 3x3x11      3x3x11       9x11        9x11

Edited March 16, 2014 by Asterisk Propernoun

[Rift enchanted 380]

I highly doubt anyone can help you here with that bro.