I’m sure you’ve probably heard the phrase, “It just clicked into place.” Last night (or rather, very early this morning) I experienced that. Literally. Like it was an actual audible “click” sound as a realized something regarding the prime numbers in the “factor field” that I’ve developed in Excel.
To give some background, I’ve been conversing with someone via e-mail after my post the other day that included my reference to the factor field. This person has looked over my spreadsheet and given some comments, and last night I responded to him. Which meant that in the process I was looking over the sheet a great deal and doing lots of mathematical conversions and the like.
Anyway, I went to bed after I sent the e-mail. And at about 12:30 in the morning, I suddenly shot up in bed because I heard the “click” as something slid into place in my brain. Yeah, I did the whole caricature thing of having the light dawn on me :-)
Of course the only problem is that there’s like maybe a dozen people on Earth who would care about my realization, and I don’t know any of them personally. But I figure why not post it into the Internet anyway? So I will.
First, I should note that with the factor field, I’ve mentioned the “spike” that occurs spaced out every 6 digits. Because of this, I wanted to see what prime numbers would look like in base-6 format (using only 0-5 for your digits, just as binary uses only 1 and 0). Last night, I compiled a short list of some of the primes and e-mailed them to the person I’ve been corresponding with, so here’s the list of primes from 2 - 101 with their corresponding base-6 conversion:
2 = 2
3 = 3
5 = 5
7 = 11
11 = 15
13 = 21
17 = 25
19 = 31
23 = 35
29 = 45
31 = 51
37 = 101
41 = 105
43 = 111
47 = 115
53 = 125
59 = 135
61 = 141
67 = 151
71 = 155
73 = 201
79 = 211
83 = 215
89 = 225
97 = 241
101 = 245
And just for fun, converting the last 10 primes on the Excel sheet gives us this:
65413 = 1222501
65419 = 1222511
65423 = 1222515
65437 = 1222541
65447 = 1222555
65449 = 1223001
65479 = 1223051
65497 = 1223121
65519 = 1223155
65521 = 1223201
So as you can see, all the prime numbers after 2 & 3 end in either 1 or 5 in base-6.
Now because my hypothesis (which I lack the mathematical skills to prove beyond a shadow of a doubt) is that all prime numbers end in 1 or 5 in base-6, as I was trying to fall asleep I thought: “At what point do the prime numbers interfere with the 6-spike?” That is, at what point on the number series do prime numbers fall either 1 above or 1 below the spike (1 above the spike corresponds to a number ending in 5, one below corresponds to a number ending in 1).
Obviously 2 and 3 are ruled out from the get-go, because 2 x 3 creates the 6-spike. So I started with 5. And here’s what I got:
In this graphic, the blue lines are the 6-spike. The red cells are those that occur either 1 above or 1 below the 6-spike. The black cells are the other cells that do not fall either one above or one below the 6-spike.
I left the factors in the cells too. As a result, trace the 5-line down and you see that the first time it falls 1 above or 1 below the 6-spike (1 above in this case) is at 5 x 1. The next time it falls 1 above or 1 below the 6-spike (1 below in this case) is at 5 x 5. The next time (1 above) is at 5 x 7. Then again (1 below) at 5 x 11. Finally, it comes 1 above at 5 x 13.
Now look at the 7 line. 7 does the exact same thing but with the above/below polarity switched! The first time it appears is 1 below at 7 x 1. Then at 1 above at 7 x 5, etc. We see the same thing with the 11 and 13 lines. Thus we have:
N = multiple of 6.
N - 1 goes in an above/below sequence.
N + 1 goes in a below/above sequence.
And the real kicker…the N +/- 1 is itself the number that the factors are based on! Thus, take a factor of 6. Subtract 1. It is now 1 below a factor of 6. Multiply by 5 (i.e. 6 -1) and you will be 1 above a factor of 6. Multiply by 7 (i.e. 6 + 1) and you will be 1 below a factor of 6. Multiply by 11 (i.e. [2 x 6] – 1) and you will be one above a factor of 6. Multiply by 13 (i.e. [2 x 6] + 1) and you will be one below a factor of 6.
Let’s give an example. 24 is a factor of 6.
24 – 1 = 23. 23 x 5 = 115. 115 – 1 = 114. 114 = 6 x 19.
24 + 1 = 25. 25 x 5 = 125. 125 + 1 = 126. 126 = 6 x 21.
24 – 1 = 23. 23 x 7 = 161. 161 +1 = 162. 162 = 6 x 27.
24 + 1 = 25. 25 x 7 = 175. 175 – 1 = 174. 174 = 6 x 29.
24 + (2 x 6) – 1 = 35. 35 x 5 = 175. 175 – 1 = 174. 174 = 6 x 29.
24 + (2 x 6) + 1 = 37. 37 x 5 = 185. 185 + 1 = 186. 186 = 6 x 31.
24 + (2 x 6) – 1 = 35. 35 x 7 = 245. 245 + 1 = 246. 246 = 6 x 41.
24 + (2 x 6) + 1 = 37. 37 x 7 = 259. 259 – 1 = 258. 258 = 6 x 43.
So, to generalize it further, let us define an N class number as a positive number that is divisible evenly by 6.
1. (Nx - 1) x (Ny - 1) = X. X – 1 is an N class number.
2. (Nx + 1) x (Ny - 1) = X. X + 1 is an N class number.
3. (Nx - 1) x (Ny + 1) = X. X + 1 is an N class number.
4. (Nx + 1) x (Ny +1) = X. X – 1 is an N class number.
To test this, let Nx = 36 and Ny = 12.
1. (36 – 1) x (12 – 1) = 35 x 11 = 385. Subtract 1 and 384 = 6 x 64.
2. (36 + 1) x (12 -1) = 37 x 11 = 407. Add 1 and 408 = 6 x 68.
3. (36 – 1) x (12 +1) = 35 x 13 = 455. Add 1 and 456 = 6 x 76.
4. (36 + 1) x (12 + 1) = 37 x 13 = 481. Subtract 1 and 480 = 6 x 80.
But we can further generalize this by creating a new class, which I will call the P class. The P class is defined as any number that is N +/- 1. So take any N class, add or subtract one from it, and that is a P class number. From the above, we therefore know that any P class multiplied by another P class number yields another P class number. It comes in the following format.
Let us define Pdown as a N – 1 class number and Pup as an N + 1 number.
1. Pdown x Pdown = Pdown.
2. Pup x Pdown = Pup.
3. Pdown x Pup = Pup.
4. Pup x Pup = Pdown.
Now my theory is that all prime numbers are P class numbers, but not all P class numbers are prime numbers. After all, since a P class x a P class yields a P class, then we have proof that P classes can exist with factors. But here’s my theory on that: the only P class numbers that are not primes are those P classes that are created by multiplying other P class variables.
In other words, when thinking about primes, one need not worry about anything other than P class integers.
Let me explain by showing the first few primes again. After 2 and 3 (which create the 6-spike in the first place) we have 5, 7, 11, 13, 17, 19. Each of these shows both sides of the 6-spike.
The first “break” occurs after 23, because 25 has factors. But what are the factors of 25? Only 5 x 5. And 5 is a prime number. In fact, 5 is the smallest prime number that comes into play (again, because 2 and 3 are working to create the 6-spike so they are irrelevant here). In fact, if we multiply the smallest relevant primes, we get:
5 x 5 = 25.
5 x 7 = 35
7 x 7 = 49
5 x 11 = 55
5 x 13 = 65
7 x 11 = 77
5 x 17 = 85
7 x 13 = 91
5 x 19 = 95
And these results are all the numbers that are missing from the 6-spike as primes.
In any case, I think it’s safe to say that we can define a prime number as any P class number that is not divisible by any other P class number. And I also think that P class number that are divisible by any numbers at all are only divisible by other P class numbers. Therefore, we need not worry about any other numbers when testing for primes.
