The Prime Number Triangle
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. |
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). (Note that because 2 is the only even prime number it has been suggested that the primes should start with 3). 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).
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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.
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The half-diffs between diagonally adjacent primes through 29
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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.
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 all the counting numbers through 142 to be systematically generated in this manner. |
My 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).
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 Duo-Number Line.
The duo-number line 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 a duo-number line (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.
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. See diagram 6.
The duo-number line 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 a duo-number line (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.
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. See diagram 6.
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 Kramer’s conjecture:
Consider the set of all prime numbers Pn where n={1,2,3…}. For every counting number A>1 there exists least one positive integer N such that
(Px - Pn)/2 = A or (Py - Pn)/2 = A
where X = N + ((8*N + 1) - 1)/2
truncated UP to the nearest integer; and
Y = X + 1.
(Px - Pn)/2 = A or (Py - Pn)/2 = A
where X = N + ((8*N + 1) - 1)/2
truncated UP to the nearest integer; and
Y = X + 1.
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 following table shows how many times a given integer occurs
in each row of the prime triangle.
in each row of the prime triangle.
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
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 among 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 among the counting numbers. Diagram 9b.