Spatial transformations of numerical sequences
Here the counting numbers are constrained within a 60 degree orthogonal grid. In this algorithm the area of the shapes starts with 2 equilateral triangles and increases by one equilateral triangle, depicting each counting number where the even area shapes originate on the vertical axis and the odd area shapes originate on the 60 degree axis and they are required to fill all space.
While the counting number sequence is totally predictable, that is non-random, the interface between the odd and even areas is no longer predictable and the progression of shapes is unpredictable. To determine the shape of the next cell requires progressing through all the preceding cells. This is similar to cellular automatons as described by Wolfram in "A New Kind of Science".
While the counting number sequence is totally predictable, that is non-random, the interface between the odd and even areas is no longer predictable and the progression of shapes is unpredictable. To determine the shape of the next cell requires progressing through all the preceding cells. This is similar to cellular automatons as described by Wolfram in "A New Kind of Science".
Prime numbers are responsive to restriction within a triangular matrix. Prime triangles position the primes in numerical order along the rows of a triangle . By taking the half-difference or "half-diff" of adjacent primes within this triangular matrix it turns out over a large enough prime triangle that all the counting numbers within a given range are generated. Here through 29 the counting numbers 2-6 and 8 are generated by this process, skipping 7.
When another row is added to the prime triangle through 47 the counting numbers 1-11 are generated capturing the missing 7. Using a computer program the prime triangle has been taken out to the first 50 million primes and has generated the first 100,000 counting numbers in this manner excluding 1. This represents a spatial transformation from prime numbers to counting numbers. A more detailed technical article is available. See below.
Prime Number Triangle
Russell Kramer
Eric Pierce
William Price
Introduction
Traditional approaches to understanding prime numbers such as the Sieve of Eratosthenes, prime number theorem and Riemann hypotheses posit the existence of counting numbers and regard “primeness” as a feature or substructure thereof. While many authors refer to prime numbers as “building blocks of integers” this is typically meant only in a multiplicative sense (i.e. in terms of prime factors). An alternate novel approach is pursued where prime numbers are used to derive the counting number system in an explicitly additive fashion. A basic requirement is that counting numbers are systematically generated from the primes. From this perspective, a new prime number conjecture is advanced.
Traditional approaches to understanding prime numbers such as the Sieve of Eratosthenes, prime number theorem and Riemann hypotheses posit the existence of counting numbers and regard “primeness” as a feature or substructure thereof. While many authors refer to prime numbers as “building blocks of integers” this is typically meant only in a multiplicative sense (i.e. in terms of prime factors). An alternate novel approach is pursued where prime numbers are used to derive the counting number system in an explicitly additive fashion. A basic requirement is that counting numbers are systematically generated from the primes. From this perspective, a new prime number conjecture is advanced.
Part 1
There are several triangles that can be categorized as “prime triangles. Diagram 1 is the first prime triangle.
The apex of the first prime triangle begins with the “first” prime number (2)*. The second row has the next two consecutive primes 3 and 5, the third row has the next three consecutive primes 7, 11 and 13, and so on. The “diffs” are the differences between diagonally adjacent primes within the prime triangle. Thus for example between the first two rows we have 3-2=1 and 5-3=2; between the next two rows we have 7-3=4, 11-3=8, 11-5=6 and 13-5=8; and so on. See diagram (2).
With the exception of the number 2, all prime numbers are odd. Hence the difference between any pair of primes excluding 2 will always be even. It is constructive to divide the diffs by 2, thus defining the set of “half-diffs” (see Diagram 3). Apart from the first two, all half-diffs are integers. We may tabulate the half-diff integers generated so far by reordering them numerically and eliminating redundancies to obtain the set {2, 3, 4, 5, 6, 8}. Note that 7 is skipped. Adding the next row to the first prime triangle yields Diagram
4.
4.
The half-diff integers derived from adding a fifth row to the first prime triangle are {7, 9, 10, 11, 12, 14}. Combining with previous results obtains the set of half-diff integers derived from the entire first prime triangle of 5 rows:
{2, 3, 4, 5, 6 ,7 ,8 ,9 ,10, 11, 12, 14}. We have now collected the counting numbers 2 through 14, excluding only 13.
{2, 3, 4, 5, 6 ,7 ,8 ,9 ,10, 11, 12, 14}. We have now collected the counting numbers 2 through 14, excluding only 13.
From here it is casually observed that as additional rows are added to the prime triangle, more and more consecutive integers are captured in the resultant set of unique half-diffs. A prime triangle with 30 rows is sufficient to permit the counting numbers through 142 to be systematically generated in this manner.
Kramer's hunch at this point was that as the prime triangle gained more and more rows it would keep capturing more and more consecutive integers and that were this process of adding rows to the prime triangle and revealing the half-diffs were to continue indefinitely all integers would be captured. An extended computer analysis has taken the prime Triangle out to 50 million primes and generated the first 100,000 consecutive counting numbers (excluding the number 1). Hence the patterns hold up to extreme high level of iterations.
The prime triangle provides a schema for selecting a finite and bounded set of prime pairs from the infinite, unbounded set of all possible prime pairs. In the prime triangle, each prime is paired with at most 4 other primes. Additionally, because the primes are paired across adjacent rows, the difference between them is strongly constrained relative to the overall size of the triangle. The probability of capturing all of the counting numbers is lessened by the elimination of most permutations of prime pairings. For instance in diagram 4 the half-diff 2 results from the pairing of 7 and 3. That is the only prime pair in the prime triangle which generates the half-diff of 2. There are numerous other prime pairings who’s half-difference generates 2 but which are not included in the prime triangle, for example {23,19}, {17,13} and {41,37}. In light of the twin prime conjecture and related theories, the number of such pairings may in fact be infinite.
Part 2
To efficiently continue the analysis, it is helpful to develop a coordinate system to encapsulate the structure of the prime triangle -namely, the Integer Triangle.
The integer triangle has the same structure as the prime triangle, but the nodes are simply indexed by the counting numbers starting with n=1. The rows are indexed by the counting numbers starting with m=1. This creates the integer triangle ( Diagram 5), where n and m both embody the counting numbers, but their relationship to each other is not synchronized; perhaps you could say they are meta-synchronized. Their alignment embodies the dichotomy between addition/subtraction and multiplication/division from which the mystery of prime numbers emerges. The numbers in the “triangular numbers” that is the integers in the form of n(n+1)/2).
Notice that for the rightmost node of each row, the quantity (8n+1) equals the squares of consecutive odd numbers starting with 3. In other words 8n+1=(2m+1)(2m+1). (Rewriting as n = ((m+ m) / 2 reveals that n is the sum of the arithmetic series derived from the regular counting progression of m.)
The construction based on successive squares is a similarity to the Ulam Spiral. In fact, by darkening the nodes where n is prime, vertical stripes appear which
are roughly analogous to the rays of the
Ulam spiral.
The construction based on successive squares is a similarity to the Ulam Spiral. In fact, by darkening the nodes where n is prime, vertical stripes appear which
are roughly analogous to the rays of the
Ulam spiral.
When the prime numbers are overlaid on the duo-number line, it is clear that each prime number has an associated index n. Let us define Pn = the prime number whose index is n. We can now formally state the Kramer-Price conjecture:
Part 3
There is more. The occurrence of integers derived from the triangle shows redundancies along with non-numerical order. Diagram 7 shows the prime triangle extended to 7 rows. Diagram 8 shows the integers (half-diffs) as they are generated. This is a generalized bin showing the total occurrence of each counting number as the rows of the prime triangle increase through row 7. The redundancies will be further reviewed later.
The following graphs show the amount of times a number is represented by half the difference of two primes in the prime triangle. The X- axis are the numbers and the Y- axis is the number of times it occurs as a half difference of two primes in the prime triangle. Diagram 9 shows the results from a triangle with 4000 rows.
X axis = a given half-diff
Y axis = the minimum row it occurs in
Note that diagram 9 shows a clearly functional relationship between the X and Y axis (i.e Y=g(X). The question arises as to what the value of g is as well as the method to derive or at least approximate this. Remembering that in essence diagram 9a shows the “distribution” of the counting numbers amongst the primes in the prime triangle as derived from the “half-diffs” it is appropriate to compare this graph with the graph of π (X), i.e. the number of primes less than x which is the distribution of primes amongst the counting numbers. Diagram 9b.
Y axis = the minimum row it occurs in
Note that diagram 9 shows a clearly functional relationship between the X and Y axis (i.e Y=g(X). The question arises as to what the value of g is as well as the method to derive or at least approximate this. Remembering that in essence diagram 9a shows the “distribution” of the counting numbers amongst the primes in the prime triangle as derived from the “half-diffs” it is appropriate to compare this graph with the graph of π (X), i.e. the number of primes less than x which is the distribution of primes amongst the counting numbers. Diagram 9b.
It is interesting to note the similar slopes of both graphs. Even down to the detail that both graphs start with a slightly steeper slope and then slightly level off. While not conclusive this may indicate that the value of “g” is related to π.
The following graphs 10a-10d represent one continuous graph in 4 sections to be viewed from left to right. The x axis = the counting numbers and the y axis = the number of times a given number (integer) occurs as a half difference in the prime triangle.
The following graphs 10a-10d represent one continuous graph in 4 sections to be viewed from left to right. The x axis = the counting numbers and the y axis = the number of times a given number (integer) occurs as a half difference in the prime triangle.
It is interesting that the number of times a given number occurs increases as the number of rows increases. Further it tends to increase more for larger numbers than for smaller ones. I guess this makes sense because the space between the primes tends to get larger and larger (not always of course).
A prime triangle can be constructed with 3 at the apex instead of 2 shifting the primes within the triangle. Many identical pairings of primes occur and new prime “diff” pairings. The new triangle apparently retains the property of systematically generating the counting numbers. To preserve this property, the largest prime which can be placed at the apex is 17. That is to say there are exactly 7 prime triangles with apexes 2, 3, 5, 7, 11 ,13 and 17. Each of these prime triangles has their own unique.
A prime triangle can be constructed with 3 at the apex instead of 2 shifting the primes within the triangle. Many identical pairings of primes occur and new prime “diff” pairings. The new triangle apparently retains the property of systematically generating the counting numbers. To preserve this property, the largest prime which can be placed at the apex is 17. That is to say there are exactly 7 prime triangles with apexes 2, 3, 5, 7, 11 ,13 and 17. Each of these prime triangles has their own unique.
There is a similarity between the K-P conjecture and Goldbach’s conjecture. Goldbach’s conjecture indicates that all even integers greater than two can be expressed as a sum of two or more primes. Kramer’s conjecture indicates that all integers can be expressed as (half) the difference between two prime numbers, selected from a finite, bounded region. There is speculation that the K-P conjecture might represent an alternative expression to Goldbach’s conjecture although a precise connection has not been clearly established.
Also from the Prime triangle and further generalizations a series of proto-prime number formulas can be derived. The K-P conjecture suggests that the prime triangle derives all the counting numbers. While it was mentioned that there were redundancies (for example the number 15 occurs as a half-diff in the Prime Triangle seven times) the redundancies were bypassed to observe other properties. The redundancies themselves are significant.
For one there appears to be a bias towards half-diffs in the Prime Triangle which are multiples of three. As well taking the half-diff of 15 and given the prime triangle and the integer triangle to index the primes:
Also from the Prime triangle and further generalizations a series of proto-prime number formulas can be derived. The K-P conjecture suggests that the prime triangle derives all the counting numbers. While it was mentioned that there were redundancies (for example the number 15 occurs as a half-diff in the Prime Triangle seven times) the redundancies were bypassed to observe other properties. The redundancies themselves are significant.
For one there appears to be a bias towards half-diffs in the Prime Triangle which are multiples of three. As well taking the half-diff of 15 and given the prime triangle and the integer triangle to index the primes:
Combining the first six elements in table 2 we have:
Equation 1 is among a vast amount of “proto-prime number formulas” which can be derived from the Prime Triangle and from further generalizations there in. These equations are highly significant. In a general sense they are formulaic and reveal relationships between the primes and the counting numbers. Previous efforts to find a function (where
f(n) =Pn where n=(1, 2, 3, 4……) had very limited success as approached higher values.
The formulaic approach to primes has generally given way to a more statistical and probabilistic approach such as is evident in the Prime Number Theorem. Further investigations into the Prime Triangle and further generalizations point to a variety of equations of the general form:
f(n) =Pn where n=(1, 2, 3, 4……) had very limited success as approached higher values.
The formulaic approach to primes has generally given way to a more statistical and probabilistic approach such as is evident in the Prime Number Theorem. Further investigations into the Prime Triangle and further generalizations point to a variety of equations of the general form:
In this case it is desired to find an extended equation of this form where the pattern of plus and minus signs is predictable opening the possibility to predict subsequent primes. In essence a prime number formula.
Summary
By starting out with the prime numbers within the framework of the prime triangle a unique approach to understanding the prime vs. counting number relationship is revealed. The novel approach of using the primes to construct the integers in an explicitly additive fashion is quite revealing. A comparison between diagrams 9a and 9b show a possible relationship between known distribution of primes within the counting numbers vs. the distribution of counting numbers within the prime triangle. A construction based on successive squares by superimposing the primes onto the integer triangle provides optimal ordering and indexing of the primes and from this the Kramer-Price conjecture is advanced. The counting numbers are derived from the ordered set of all primes using a finite logical method.
Further research will pursue more refined estimates of g(X) in diagram 9a. Other prime triangles with different apexes (3, 5, 7, 11, 13 and 17) will be analyzed and results cross referenced. The analysis of the redundant “half-diffs” in the Prime Triangle yields a variety of proto-prime number formulas in turn revealing further details on the relation between the primes and counting numbers and reviving the formulaic approach to investigating prime numbers.
By starting out with the prime numbers within the framework of the prime triangle a unique approach to understanding the prime vs. counting number relationship is revealed. The novel approach of using the primes to construct the integers in an explicitly additive fashion is quite revealing. A comparison between diagrams 9a and 9b show a possible relationship between known distribution of primes within the counting numbers vs. the distribution of counting numbers within the prime triangle. A construction based on successive squares by superimposing the primes onto the integer triangle provides optimal ordering and indexing of the primes and from this the Kramer-Price conjecture is advanced. The counting numbers are derived from the ordered set of all primes using a finite logical method.
Further research will pursue more refined estimates of g(X) in diagram 9a. Other prime triangles with different apexes (3, 5, 7, 11, 13 and 17) will be analyzed and results cross referenced. The analysis of the redundant “half-diffs” in the Prime Triangle yields a variety of proto-prime number formulas in turn revealing further details on the relation between the primes and counting numbers and reviving the formulaic approach to investigating prime numbers.