Continued from previous post ...
Now, before we continue, there's something that needs to be cleared up.
In my opening post I mentioned that by convention, the number 1 was declared as NOT a prime even though like other members of the prime family, it can
only be divided by itself.
Also, number 2 is unique in all the infinite primes because it's the ONLY conventional prime that is an even number ... every other prime is odd.
So, according to "convention", the 1st two primes are the numbers 2 and 3.
But if you refer back to Diagram 2, you'll notice that firstly, the number 1 DOES appear on a prime number ray (ray1) and secondly, that neither the
number 2 or the number 3 appear on any of the 8 rays associated with the infinite amount of prime numbers.
We therefore seem to have a bit of a "conflict" here between what conventional prime number theory and my version have to say about these 1st three
prime numbers (i.e. numbers 1, 2 and 3).
Considering that I've managed to generate a geometrical model that successfully demonstrated a type of prime number distribution based on a very
simple rule that primes are generated around a base of 24, I'm prepared to go out on a limb and state that in my opinion (ONLY !!), that the number 1
should be reinstated as a prime and further that the numbers 2 and 3 should not be considered "true" primes as they do NOT fall on any of the rays
that the remaining infinite number of primes do fall on.
Also, we should be wary of considering the number 2 as prime because of it's unique status of being the ONLY conventional prime that is EVEN ... one
should rightly question why this single conventional prime should have such an "elevated and special" position compared to the infinitude of
Coincidentally, nature may even have "encoded" a clue pointing to this "base of 24" that all my work revolves around, in the three numbers located
between my 1st prime number (1) and my 2nd prime number (5). Between these two primes lie the numbers 2, 3 and 4.
The product of these three numbers (2 x 3 x 4) just happens to be 24
Now, that certainly appeals to my sense of order !
By the way, the maths and my diagrams being based on 24 is known as Modular maths ... or sometimes referred to as "clock arithmetic" ... introduced
by the mathematician Carl Friedrich Gauss in 1801.
In summary therefore, and as far as I can see, the initial five primes may actually be 1, 5, 7, 11, 13 and NOT the conventional 1, 2, 3, 5 and 7.
Ok, having got that little bit out of the way, let's take a look at other aspects of assuming that primes and their properties revolve around a base
Take a look now at the following diagram.
The numbers you're seeing on ray1 are the squares of the 1st 5 primes.
Perhaps yet another indicator that numbers 2 and 3 should NOT be considered "true" primes as I've explained above.
In fact, no matter which prime you select, when you square it, the resulting value will ALWAYS appear on ray1.
Lets use the following prime numbers to illustrate this effect ... 41, 1009 and 10007.
1. 41 x 41 = 1681
2. 1681 / 24 = 70.0416666666666 (repeating)
3. 70 + 1 = 72
4. 0.0416666666666 x 24 = 1
Therefore, 1681 (41 squared) is on Circle 72, Ray1
1. 1009 x 1009 = 1,018,081
2. 1018081 / 24 = 42420.0416666666666 (repeating)
3. 42420 + 1 = 42421
4. 0.0416666666666 x 24 = 1
Therefore, 1,018,081 (1009 squared) is on Circle, 42421, Ray1
1. 10007 x 10007 = 100,140,049
2. 100140049 / 24 = 4172502.0416666666666 (repeating)
3. 4172502 + 1 = 4172503
4. 0.0416666666666 x 24 = 1
Therefore, 100,140,049 (10,007 squared) is on Circle 4172503, Ray1
Now, go ahead ... pick some primes of your own ... square them and confirm that the squared value ALWAYS falls on Ray1.
In fact, this property of always finding the square of any prime on Ray1 is also a property of the "square of the product of two primes".
Yeah, I know ... lets try that again in English !
If you pick ANY two primes, say 7 and 11 then multiply them to get 77 ... if you then square this value of 77 to get 5929 ... once again this answer
(5929) falls on Ray1.
Having the answer falling on Ray1 will ALWAYS happen no matter which two prime numbers you choose ... amazing, huh ?
Lets try this with another two primes, say 601 and 1248007.
1. 601 x 1248007 = 750,052,207 (product)
2. 750052207 x 750052207 = 562,578,313,225,570,849 (squared)
3. 562578313225570849 / 24 = 23440763051065452.0416666666666 (repeating)
4. 23440763051065452 + 1 = 23440763051065453
5. 0.0416666666666 x 24 = 1
Therefore, the square of the product of 601 and 1248007 is on Circle 23440763051065453, Ray1
Again, pick a few prime pairs yourself and confirm that the answer ALWAYS lies on Ray1.
Once we're aware of this property attached to squaring a prime and finding the answer ALWAYS on Ray1, we can generalize and now state that the
difference between ANY two primes that have been squared will ALWAYS be a multiple of 24.
What am I saying here ?
If you take ANY prime and square it, the answer will be somewhere on Ray1.
If you take ANY second prime and square it, the answer again will be somewhere on Ray1.
If you now subtract the smaller of the two answers from the larger of the two answers, the difference will ALWAYS be a multiple of the number 24.
As an example, lets use primes 5 and 13.
5 x 5 = 25 (squared)
13 x 13 = 169 (squared)
169 - 25 = 144 (difference)
Therefore, 144 = 24 x 6 = 24n
(where n = 6)
Another example using primes 71 and 1283.
71 x 71 = 5041 (squared)
1283 x 1283 = 1646089 (squared)
1646089 - 5041 = 1641048 (difference)
Therefore, 1641048 = 24 x 68377 = 24n
(where n = 68377)
Continued next post ...
As should be very obvious by now, the primes and their properties all revolve around and are completely dependent on the value 24 !!