Herein, let “numbers” mean Natural Numbers.

Most frequently, numbers are regarded as being a string of digits in a single-file starting at zero and stretching into infinity; each digit regarded as a unique element representing either Cardinality or Ordinality. While this is not wrong, it is essentially one-dimensional. There is another way to look at numbers.

Frequentiality is the idea that each number is actually an infinite set of identical elements that exist at a given frequency on a number line.

Imagine you are sitting in a car, in the turn lane and there are a number of cars ahead of you, each with its blinker on. You can see all the blinkers including your own. You notice that mostly the blinkers are out of phase because each one flashes at a unique frequency. For this same reason, sometimes some of them blink together and if let go for long enough, all of them will blink simultaneously if only for a short time.

Like automobile turn signals, when compared to each other, numbers as frequencies can be “out of phase” and other times they exist together at a single point. It is in these moments of coexistence that this paper hopes to shed further light in the area of number theory.

The intent here is to explain only the fundamentals of Frequential Numbers. The author readily admits his relative lack of training in mathematics but views it as not only a weakness but a strength. It was a lack of training that allowed numbers to be seen as something new. If the idea of Frequential Numbers is actually something that proves to be useful, hopefully, smarter people will take it and run with it.

**Definition**

*Abstract Number Line:* An Abstract Number Line is a string of points starting from some Origin and continuing in one direction infinitely. Each point is equidistant from any adjacent points (though the actual distance is unspecified). No point has any value, cardinality or ordinality.

*Figure 1. The Abstract Number Line*

Definition

*Origin*: The point from which an Abstract Number Line originates. It is notated as a zero with a diagonal line through it. (Please see Figure 1 above)

**Axiom**

The set containing all points on an Abstract Number Line is denumerable

**Axiom**

The set containing all Abstract Number Lines is denumerable.

**Definition**

*Frequential Number*: A Frequential Number is the means by which to quantify the distance between points on an Abstract Number Line. Frequential Numbers may be any of the Natural Numbers.

*Figure 2. The distance between points on the Abstract Number Line below has been quantified as 2*

*Frequency*: An Abstract Number Line that has had the distance between its points quantified by a Frequential Number. Figure 2 above shows the Frequency 2.

**Definition**:

*Frequential Elemen*t: Any of the points of a Frequency. Though identical, each Frequential Element is a unique thing in and of itself.

**Axiom**

There is at least one Frequency for each Natural Number.

**Axiom**

The set containing all Frequencies has a denumerable cardinality.

**Definition**

*Stacked Frequencies*: A comparison between two or more Frequencies by aligning their origins and “stacking” one on top of another.

*Figure 3. The Frequencies 1, 2 and 3 have been Stacked. *

**Axiom**

Any and all Frequencies can be stacked.

**Axiom**

The Abstract Number Line is identical to the Frequency 1, and therefore we draw a 1 to 1 correspondence with the set of Natural Numbers.

*Figure 4. Frequency 1 paired with N.*

**Axiom**

All Frequencies can be compared to N.

*Figure 5. All Frequencies stacked, with their Origins aligned, and compared to N. *

**Definition**

*Power Table or PT*: The table comparing all Frequencies with all Natural Numbers.

**Axiom**

The Power Table continues upward and to the right infinitely.

**Axiom**

In the Power Table, any element aligned vertically with a Natural Number is a divisor of that number.

Example: In Figure 5 the Natural Number 12 (horizontally) has Frequential Elements from Frequencies 1, 2, 3, 4, 6 and 12 aligned vertically above it. These are all divisors of 12.

**Axiom**

In the Power Table, each Natural Number is a multiple of any Frequential Element aligned vertically with it.

**Axiom**

In the Power Table, only Frequential Elements from Frequencies of 1 and the number itself align vertically with Prime Numbers.

**Axiom**

With the exception of 2 and 3, Prime Numbers may only appear adjacent to multiples of 6 (positions of 6n-1 or 6n+1). All other numbers are divisible by 2 and 3.

**Definition**

P: The set containing all Prime Numbers.

**Definition**

*Prime Center*: The “n” in (6n + 1) and (6n – 1). These are the “centers” around which members of P are distributed.

**Definition**

*Super Prime*: 2 and 3 are Super Prime. They are not in a position of 6n-1 or 6n +1 so we will put them in a separate category and leave them there for now. From now on, the term Prime does not include 2 or 3.

**Definition**

*Primary Origin or Op*: The Origin on the Power Table with which all Frequencies align.

**Definition**

*Subsequent Origin or Os*: Any point, other than Op, at which any two or more Frequential Elements align.

**Definition**

*Possible Primes*: because Prime Numbers only appear at a position of 6n + or – 1, any number in such a position is possibly prime. Conversely, numbers in any other position are Impossibly Prime. A Possible Prime is any number 6n + or – 1.

**Definition**

*Pp*: The set containing all Possible Primes. The cardinality of Pp is denumerable.

**Definition**

*Non-Prime*: a member of the set Pp that is not a Prime Number.

**Definition**

*Pn*: The set containing all Non-Prime Numbers. Pn is a subset of Pp.

**Axiom**

Impossible Primes are not members of Pn.

**Axiom**

P is a subset of Pp.

**Axiom**

Each member of Pp is also either a member of Pn or P but not both.

In terms of Prime Numbers, the Power Table contains a lot of redundant information. It is easily condensed.

*Figure 6. The Power Table condensed to exclude information non essential to Prime Numbers.*

Horizontally, Figure 6 lists all Prime Centers with each associated Possible Prime above and below.

Example: 77 = (6n-1) where n = 13. See Figure 6 above.

**Definition**

*Condensed Power Table or PTc:* The condensed version of the Power Table.

In the Condensed Power Table, how are members of Pp determined to be members of Pn or P?

*Figure 7. The Condensed Power Table is structured. Below, all multiples of 5 are crossed off. Interestingly, these lines intersect only Prime Centers that are also multiples of 5.*

The question becomes: what happens if the pattern is extended to the left?

*Figure 8. Pattern is extende to the left.*

In order to preserve the pattern, numbers to the left of the Primary Origin are an inverted mirror of those on the right. But does this help? Does this mean that connecting like numbers across the Primary Origin establishes the pattern for the next set of lines to cross off multiples of that number?

*Figure 9. Starting at 7 and moving right.*

It turns out, yes, a line connecting two like numbers across the Primary Origin establishes the pattern for crossing off multiples of that number.

Does it work for any number?

*Figure 10. Crossing off 11.*

These red lines are determining that certain members of Pp are also members of Pn. They are themselves a kind of Frequential Element. They are established by drawing a line from a member of P, through the Primary Origin. Once established, a Determinator repeats at a frequency equivalent to the member of P that established it.

**Definition**

*Determinator*: A Frequential Element used to determine which members of Pp are also members of Pn.

*Figure 11. 5 and 7 Determinators working together*

**Axiom**

Each Determinator creates a rigidly structured pattern of members of Pp that are also members of Pn.

In looking at Figure 11 above, it becomes clear that when Determinators of different Frequencies are overlapped on the Condensed Power Table, the actual distribution of Prime Numbers becomes apparent.

Figure 11 above demonstrates that 5,7,11,13,17,19,23,29,31,37,41,43 and 47 are not determined to be members of Pn by Determinators 5 or 7 and are therefore members of P. These numbers are in fact all Prime.

The 5 and 7 Determinators are only effective to a point. Certainly 121 (20*6+1) is not a member of P? This is where the 11 Determinator will take over and crossed off 121.

**Axiom**

It is only possible to ever have 2 consecutive sets of “Twin Primes”. See Figure 11 above.

**Axiom**

The first unique member of Pp that a Determinator determines to also be a member of Pn is the square of the Prime the Determinator is established by.

**Example**

The first member of Pp that the 5 Determinator determines to be a member of Pn is 25, the square of 5. The first unique member of Pp that the 7 Determinator determines to be a member of Pn is 49. While 7 does determine 35 to be a member of Pn, it is not unique, the 5 Determinator has already determined it is a member of Pn.

**Axiom**

Determinators are classified as either useful or non-useful. Non-useful Determinators are established by members of Pn. The 25 Determinator is the first non-useful Determinator. Any member of Pp that it determines to be a member of Pn will have previously been determined by the 5 Determinator. See Figure 12 below.

*Figure 12. 25 is the first non-useful Determinator*.

**Axiom**

Like members of a Frequency, Determinators also align at Subsequent Origins. This is true for any set of Determinators.

*Figure 13. Determinators 5, 7 and 11 align at Prime Center 385.*

Figure 13 above shows the 5, 7 and 11 Determinators aligning at a Subsequent Origin.

**Axiom**

All members of a subset of N can be multiplied together to create a Subsequent Origin for that subset. If the subset represents Determinators, all Determinators in that set align at the Subsequent Origin.

*Figure 14. All Determinators aligning at the nth Subsequent Origin.*

**Theorem**

*The gap between one Prime Number and the next can be infinite.*

**Proof**

*All members of any subset of Determinators will eventually align at the same Sunsequent Origin.*

**Axiom**

There is no subset of Determinators that can cross off all members of Pp.

**Axiom**

In order for P to have a denumerable cardinality, there can be no sunset of Pp that can cross off all members of Pp.

**Theorem**

P has a denumerable cardinality.

**Proof**

There is no subset of Determinators that can cross off all members of Pp.

(Borrowing from Euclid) Assuming Figure 11 above includes all Determinators, x and y are either Prime or they are not. If either is not Prime, then our set is missing a useful Determinator which means a Prime is missing from our list. If either is Prime then our list of Primes is incomplete. This goes on infinitely.

**Axiom**

The distribution of P along N is determined by an infinite number of unique, overlapping patterns that together create an infinitely changing, apparently random pattern of distribution of Prime Numbers.

*Figure 15.*

Figure 15 illustrates that the pattern of distribution is changed when Determinators begin moving members of Pp to Pn. The numbers on the bottom represent Prime Centers while the numbers on the left represent Determinators. The pattern of distribution of members of P remains unchanged until 4, where the 5 Determinator takes over at 25. It then remains unchanged until 7 takes over at 49. It then remains unchanged until 11 takes over at 121. To complicate matters, some Determinators are non-useful which means some anticipated changes never occurr.

This makes it impossible to predict the distribution of Prime Numbers.

You can’t know where Prime Numbers exist without first knowing where Prime Numbers exist.

Asking about the infinitude of “Twin Primes” is asking a question of the distribution of Prime Numbers.