On the Concept of Representations of Prime Numbers and Prime Products

The author used to think that the only representation of prime numbers, etc. was the prime number double helix representation. However, in 2011, the author published a paper which puzzled him and it eventually dawned upon him that there was more than one representation of prime numbers. The second representation will be referred to as the hyperbolic representation. The third representation will be called the parabolic representation and is related to the hyperbolic representation. The fourth representation is the triangular representation and is related to the hyperbolic representation. Both of the primary representations, the double helix and the hyperbolic both reject that 2 and 3 are prime numbers. The hyperbolic representation is shown to be related to Lorentz-like transformations.


Introduction
The prime numbers and prime products seem to be somewhat of a changeling or perhaps more like the elephant described by the proverbial blind men, each giving an incomplete description by examining the tail, the feet, the trunk, and the ear.The author was puzzled that so many concepts of prime numbers and prime products exist around the world.Whether they are random or form a pattern, do they obey a spiral, wave (Hahn, H. K., 2008), straight line, etc.In order to account for these varied ideas, the author thinks that prime numbers and prime products can be found in many different so-called representations.The author thinks that he has found four such representations.

The Double Helix Representation (Bissonnet, P., 2006)
The reader is asked to imagine an infinite row, seven column array.Further, imagine that all of the non-prime numbers have been blanked out.One ends up with a set of double lines of prime numbers extending from column 1 to column 6, except for row 1 in which the number 7 is in column 7 (see Fig. 1).The blanked-out cells along the double lines are either simple prime products or multiple prime products.6s -1 (H 1 or helix 1) describes the numbers along the top line, while 6s + 1 (H 2 or helix 2) describes the numbers along the second line of each pair of lines, with s an integer and 1 s  (see Table 1).The lines can extend up into column 7, but only the simple and multiple prime products involving 7 will appear.The multiplication rule that is obeyed is the following: where m, n and r are integers 1  , and a, b, and c are +1 or -1.Another rule of the multiplication is that

The Hyperbolic Representation
The reason as to why this is given the attribute of 'hyperbolic' will be given later in this paper.Consider two integers D and m, with D > m, and such that the smallest integer which will allow an equality consistent with D > m is D ≥ m +1.If D is an odd number, them m is even and if D is an even number, then m is an odd number.In addition, one or the other of D or m must be divisible by 3, but not both.Now consider numbers P 1 and P 2 defined as: PP N  which could be a prime number, a simple prime product, or a multiple prime product 1 P can be a prime number, a simple prime product, or a multiple prime product.
The Mathematician, 2010) Examples are given in following Table 2:   Table 2

The Parabolic Representation
The parabolic representation is related to the hyperbolic representation.

Figure 2. Parabolic representation
This is a parabola with solutions for x at y = 0.One notices that the supposed prime numbers 2 and 3 are conspicuously absent from both representations.Let us consider first the hyperbolic representation in order to investigate the products of 2 and 3 with other prime numbers and prime products:

The Triangular Representation
Table 3: Values of D and m using hyperbolic method for numbers 2 and 3 It is quite apparent from the above Table 3 that (1) the fractions violate the postulate that D and m are integers and (2) the entries which are indeed integer, one being odd and the other even, violate the postulate that one or the other must be divisible by 3. Therefore, the hyperbolic representation does not accept that 2 and 3 are prime numbers.
Let us now consider the double helix representation.The comparison of 2 and 3 with the prime numbers beginning with 5 on Helix 1(H 1 ) and 7 on Helix 2 (H 2 ) results in the following short tables of indicators that 2 and 3 are not true prime numbers: 1 s  .As shown in a previous paper (Bissonnet, P., 2017), s is a composite number which is equal to the sum of two other integers: s = r + n, upon referring to Fig. 1, r is the row number and n is the complex of double lines, the first being n = 0, then n = 1, etc. S is definitely an integer with the minimum value of 1.Therefore, the double helix representation also rejects 2 and 3 as being true prime numbers.

The Logic of 2 and 3 Not Being Prime Numbers
If the double helix representation and the hyperbolic representation are true portals through which to observe prime numbers, simple prime products, and multiple prime products, then is it not a somewhat unsound and circular type of reasoning to assume that 2 and 3 are prime numbers, when such numbers show up in the very definitions/postulates of what is attempting to be proven or ascertained.Namely, the author is referring to the mention of 6 in 6s -1 and 6s + 1.We know that 6 is 2 times 3. What about the hyperbolic representation, where one of the requirements is that one or the other of D or m must be divisible by 3? The author believes that this is further mounting evidence for 2 and 3 not being true prime numbers.

Conclusion (Bissonnet, P., 2011; Bissonnet, P., 2016)
It is very disturbing that both of the primary representations (double helix and hyperbolic) seem to view prime numbers, as well as simple prime products and multiple prime products, on the same or equivalent mathematical footing.It is the dream of mathematicians to somehow separate or filter all prime numbers from the infinite variety of numbers.The current trend is to push further the goal of solving the Riemann Hypothesis.This author sees two very troubling difficulties to this goal; namely, as just stated (1) the seemingly equal mathematical footing given by nature to prime numbers and their products, and (2) the budding recognition (at least by this author) that 2 and 3 are not true prime numbers.In physics, there is a principle that tells an experimenter that there is a limit to how much information he can extract from a system, namely: the uncertainty principle.Whether or not there is a similar principle at work in mathematics is unknown, but it pays to be cautious just the same.
The hyperbolic representation was given the name of 'hyperbolic' for two reasons.
(1) Because of the following argument.Consider the form of If we divide both sides by N, we obtain This is an important equation, because we recall from relativity theory the equation Could this imply that prime products might indicate a way that special relativity can be united with quantum mechanics, since quantum mechanics deals with integers in many instances, and this does seem to exemplify a way that integers

Fig. 3
Fig. 3 illustrates the essentials below.The line of numbers starting with 25, 36, 49, etc. is referred to by this author as the backbone, with yellow and turquoise ribs extending from it to numbers located in various cells.The numbers in these cells are the values of N (the prime product) for various prime numbers.The triangular representation is related to the hyperbolic representation.Just as an introduction, (use the zoom function on your computer to view Fig. 3 or click Excel spreadsheet icon), find the number 253 which is the product of 11 and 23.The number 253 sits on top of the dashed line m = 6.The dashed lines represent H 2 or 6s +1, while the solid lines represent H 1 or 6s -1.Notice how the dashed lines are spaced after every two solid lines, possibly indicating a regularity.

Figure 3 .Figure 4 .Figure
Figure 3. Exhibits relation of m's, H 1 , H 2 , prime numbers, and prime products calibration hyperbola curves in special relativity.This means that, strangely, the values of D of transformations similar to the Lorentz transformations, although the β or velocity factor has little meaning.
Seven column array in which prime numbers and prime products arrange themselves . Various D and m values for various prime numbers and prime products

Table 4 .
Shows that s is not an integer for prime numbers multiplied by 2 or 3