(by Stefano Maruelli: Note I'm nota a professor, just an enthusiast watcher of this interestinf field ! )
This is my little opinion on...
0) WHAT ARE NUMBERS ?
An integer "N" can be written as a simple number (F.e. 35...) or divided as products of it's factor ( 5 x 7 ). Primes number are the a few numbers of Numbers that can build all the infinite rest of all the numbers.
But many secrets of the Numbers rest, at the moment, not discovered. If you start to work with numbers you will soon understand that prime numbers can be well rapresented by Vectors, of known module, but of unknown direction.
1) WHAT ARE PRIMES
?
Primes (an their powers) can be
rapresented as base of an hypervolume that describe all the Integer
numbers: any integer can be written as a combination of 1 or more primes
numbers or of their power(s).
Each prime (vector) is orthogonal to the other one.
These means that is easy to build an hypervolume having on the X,Y,Z, K........infinite axis based on prime number.
In this simple hypervolume only PRIMES (and their power) are single points on their axis.
In this hypervolume if the number you are studing is
rapresented by an area or a (hyper) voulme it signify that this numberis not a
prime. (Trivial)
The Hipercube with x= 2^a , y= 3^b , z= 5^c primes number axis
2,3,5 and their 2^a , 3^b , 5^c powers are points on this hipervolume.
Any other non prime numbers is a combination of 2^a*3^b*5^c where a,b,c are integer from 0 to infinite...
For example in this cube: the number 150 is a volume due to = 5^2*3^1*2^1
For example in this cube: the number 8 is a point due to = 2^3
For example in this cube: the number 60466176 is an area due to = 2^10*3^10
For example in this cube: the number 5,094...*10^14 is a volume due to = 2^10*3^10*5^10 (etc...)In the 4 dimension ipervolume 7^d will add another dimension; in 5 dimension 11^e etc....
2) PHISICAL METHOD TO DISCOVER IMMEDIATELY IF A "Nx" IS A PRIME:
If we have an hyper-container full of hyper-liquid and we immerse a number in it, we can have 2 cases:
1- The liquid will not comes out from the container: so our number do not has volume then it cane be:
A) a PRIME or its Power (as told this are just
points)
B) a product of 2 primes (coprime), or their power
(as told rapresented by an area)
2- The liquid comes out from the container: so our number has volume and for sure is not a prime.
3) HOW TO DETERMINE ALL THE
PRIMES SEQUENCE:
There are many function that gives all the primes numbers:
mine is :
N(X) is and
odd number
N2 is an integer between 2 and N(x)-1
If n(x) is a
prime number the F1 result is =0.
What is not funny is that is not
possibile to find the continous function that gives all the zero
of this F1 function.....
This function is nothing more than the work (brute force) of keep a number N(X) an try to divide it by any other littlest number... If one divisor was found the function gives a result that is not zero (numbers of divisor)
Turn the problem as you want.. I think is time to declare as axiom that primes are the perfect RANDOM distribution.... no way to know who will be the pi+1sequence numbers exactly... every time we need to make a "primary test"
5) TETRA PRIMES
TWINS:
After some year of little investigation from any face I just discover in some point of view of the problem that there are many primes that are onto the: 6m+1 line, many twins prime onto the 6m+/-1 lines and many double triple etc... twins. Many TETRA primes are on the y= (x-1)^2*(X-b)^ form (where x= 6m and m= 2,3,5,7)... etc...
SOME EXAMPLE OF TETRA PRIMES TWINS:
12-5 = 7
12-1 = 11
12+1 = 13
12+5=
17
18-5 = 13
18-1 = 17
18+1 = 19
18+5=
23
42-5 = 37
42-1 = 41
42+1 = 43
42+5=
47
(6x5x3) 90-7= 83
(6x5x3) 90-1=
89
(6x5x3) 90+1= 91
(6x5x3) 90+7= 97
Prof. DiNoto and his team prove that the TETRA PRIME are infinite. Probably there are Esa Twins Primes etc...
6) THE SHAPE OF THE PRIMES (3n+1 and 3n+2 forms):
In the following picture you can see how Primes can be drawn (one way of many)
7) FUNNY CONCERNINGS:
ALL the Powers of Odds numbers are in the fomr of 3n+1
Is trivial to demonstrate it.
8) ANOTHER IMMEDIATE METHOD (Compass method) TO CHECK IF "N" IS A PRIME OR NOT:
If you have an odd integer and you're not sure if it's a prime or not you can have an immediate answer doing this trick:
- Immagine that your numbers "N" is a defined radio source emitter point fixed onto the X axis.
- Bring a sure PRIME number "P" little than "N"
- Fix it onto the X axis
Now immagine that:
P is an emitter of a wave of "P" half period
N is an emitter of a wave of "N" half period
The closest points where the P and N waves will has the same phase are on the circle at P x N radius.
So if you write 2 circles with a kompass with centre in P and N
with P x N radius you'll immediately found at their the
intersection "C"
point that has Xc and Yc coordinates.
The trick: if N is a prime, the Yc coordinate is the minimum distance from X axis, and Xc is EVER at:
P+ 1/2 * (N-P).
If N is not a prime: will be clear that N is not a source
point and the real intersection will be (much) lower than Yc
and Xc will
not be in the middle from P and N.
For example if you keep N=9 and P=5 is clear that 9 is not a prime, so not a
source, ans at the radius 5*9=45 you'll find
an intersection with
Xc :
Xc = P+ 1/2 * (N-P) equal to 5+ 1/2*(9-5) = 7
but you can also find an intersection at at the lower radius 3*5=15 and a minimum point at:
X'c = P+ 1/2 * (N'-P) equal to 3+ 1/2*(5-3) = 4 with lower Y'c
If there will be a relation is more clear that all the points related to Prime numbers will have an X'C in the form:
X'c = P+ 1/2 * (N'-P) where the 1/2 is the Reemann critical line, and is EVER 1/2
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