The twin prime counting function and it s consequences

Transcription

1 The twin prime counting function and it s consequences Chris De Corte chrisdecorte@yahoo.com October 11,

2 The goal of this document is to share with the mathematical community the results of my attempts to create a twin counting formula. I tried many different regression models first but the outcome of none of them satisfied me. Therefore, I wanted to use the same methodology that I used to derive my standard prime counting function to derive a twin prime counting function. I believed that I have succeeded in this and published the first version of this document on March 3, Due to a comment from someone, I recently changed a constant that I used in my original document to a constant that came back in my (single) prime counting formula. Now the pieces seem to be one step nearer to being in place. 2

3 CONTENTS CONTENTS Contents 1 Key-Words 4 2 Derivation of the formulas Forming the basis Main Part The twin counting formula Calculations and testing Microsoft Excel C Pari/GP Consequences About the correction factor About the number of twin primes Maximum distance between twin primes References 8 6 Figures 9 3

4 2 DERIVATION OF THE FORMULAS 1 Key-Words Prime numbers, number theory, twin prime, conjecture, zahlentheorie, Brun, Hardy-Littlewood, Polignac, Yitang Zhang; [1]. 2 Derivation of the formulas 2.1 Forming the basis Suppose an empty x-axis with all the natural numbers still vacant. We can ask ourselves what the chance would be, at the starting point, that twin primes could be created. We can assume that this will be 100% or 1. Therefore, we introduce the first prime 2 by adding a sinus wave with a period of 2 (figure 1). If we assume that all the numbers where this wave goes through are crossed out as twin prime candidates, then we can say that the chance to find twin primes suddenly diminishes with 1/2. It is clear that the couples (3,5), (5,7),... can still be twin primes. So the new twin prime probability becomes 1 1/2 = 1/ Main Part We now introduce the second prime by adding a sinus wave with a period of 3 (figure 2). It is clear that the number of possible twins is diminishing again but the question is with how much? Since the only position (phase) when this latest sinus is not diminishing the number of twin primes is when it crosses the x-axis on the same time as the sinus wave of period 2. This means that this wave will reduce the likelihood for new twins with (3-2)/3. So the new twin prime probability becomes 1/2 1/3 = We now introduce the third prime by adding a sinus wave with a period of 5 (figure 3). This wave will reduce the number of twins during 3 of his 5 possible random phases. This means that this wave will reduce the likelihood for new twins with (5-2)/5. So the new twin prime probability becomes /5 = 0.1. The reader can test the next step using figure The twin counting formula We continue the same logic for the other prime numbers and derive a general formula: prob i = prob i 1. p i 2 p i (1) 4

5 2.3 The twin counting formula 2 DERIVATION OF THE FORMULAS In the previous we were still calculating individual probabilities. These were only valid immediately after a new prime and before the next new prime. To know the total probability of the occurrence of a twin prime, after a set of primes already in place, we have to integrate the above formula over the length that it is valid: totprob x = prob i.(p i p i 1 ) (2) p i x We tested the above logic for a larger set of data and found that our assumption holds as will be demonstrated later. We now declare our twin prime counting formula and give it the symbol Π 2 : x Π 2 (x 3) = α p i 2 ( p j 2 ) (p i p i 1 ) (3) p j With: p i =3 p j =3 α (4) The new reader should know that in previous versions of this document, I have used π or π in stead of α 2 as the correction factor. The reason why I changed this recently should become obvious in the next paragraphs. The 1+ between the brackets of formula (3) can easily be neglected and therefore, we can rewrite our formula as: Π 2 (x 3) 1 2 α2 x p i =3 (p i p i 1 ) p i p j =3 (1 2 ) (5) p j This formula then corresponds to starting the logic of the probabilities after introducing 2 as a prime (subsection Main Part ) and not starting from an empty x-axis (subsection Forming the basis ). The above formula can be seen as an integration by parts and can hence be rewritten as follows: Π 2 (x 3) 1 2 α2 x 3 x=p i p j =3 (1 2 p j ) dx (6) There are remarkable similarities and differences between the above twin prime counting formula and the probabilistic (single) prime counting formula [2]: Π(x 2) α x 2 x=p i p j =2 (1 1 p j ) dx (7) 5

6 3 CALCULATIONS AND TESTING I have been reminded recently by a person calling himself Brian McCabe that there is a relation with the Euler-Mascheroni constant for the twin prime counting formula so that I could rewrite formula (5) to formula (6) which is a similar form as formula (7), using: α = e γ where γ is the Euler Mascheroni constant (8) 3 Calculations and testing 3.1 Microsoft Excel We first tested the above formula (3) for a limited set (until twin 15,485,651) using Excel (see figure 7). In the Excel calculation we still used π = π as correction factor. As an extra, we have calculated (in Excel) the distance between the twin primes and averaged them in buckets of 100. We have put the results in a graph and let excel calculate a trend line. The results can be seen in figure C++ Afterwards, we developed our own C++ program that generated 1000 files of about 1 million lines each. A summary of the results can be found in table 1. Using this program, we tested the above formula up to twin prime 999,999,191, which is number 3,424,506 on the list of twin primes, we come to the conclusion that, for this range, we have to correct our results with a factor of approximately We call this correction factor π. The results of the C++ program are summarized in a graph in figure 5. We added an additional line to the graph using π as a correction factor. This can be seen in figure Pari/GP First we choose the prime count up to where we want to calculate. In the example, we will calculate to prime count 530 which corresponds with prime 3821 which is number 100 of the twin primes (see Table 1). The formula 3 can be tested in Pari/GP if wanted. The code to use is:? x=530 %1 = 530 6

7 4 CONSEQUENCES? (sum(pi=2,x,0.5*prod(pj=2,pi,(1-2/prime(pj)),)*(prime(pi)-prime(pi-1)),)+1)* %2 = Or by using the simplified formula 4:? x=530 %3 = 530? Pi/2*sum(pi=2,x,prod(pj=2,pi,(1-2/prime(pj)),)*(prime(pi)-prime(pi-1)),) %4 = In Pari/GP, the maximum value for x will be as the numbers of primes is limited. However for x > 1000, this calculation becomes increasingly too challenging for the program. 4 Consequences 4.1 About the correction factor Up to now, we can not explain where the correction factor in formula (3) comes from. But it seems that the same correction factor plays a role in both the (single) prime counting formula and in the twin prime counting formula. Furthermore are there remarkable similarities and differences between both formula s which don t seem to be abnormal for mathematicians. Of coarse, both formula s are deducted using a similar (but different) logic. The correction factor might be explained by comparing the prime number theorem with the Mertens theorem but that is only an explanation by comparing formula s. Still, no physical intuitive explanation justifies this value. 4.2 About the number of twin primes In our Excel (figure 7), we can theoretically add an infinitude of additional lines each covering the last new prime. Our probability for a new twin prime will never become 0 in formula 1. Hence we can say that based on the above results, we will expect an infinitude of twin primes or that formula 1 proves the twin prime conjecture. 4.3 Maximum distance between twin primes As our calculation expands with the number of primes, our probability to find a next twin prime will diminish with formula 1. As there will never come an end to the number of primes, 7

8 5 REFERENCES the average distance will keep on increasing. This finding seems to be confirmed by the general trend in figure 8 and by the trend line formula displayed in it. So, the maximum distance between twin primes will be unbounded. 5 References 1. en.wikipedia.org ; Twin Prime ; Probabilistic approach to prime counting. 8

9 6 FIGURES 6 Figures Figure 1: After introducing the first prime 2, it seems that the number of possible twins diminished with 1/2. Figure 2: This new wave will reduce the number of twin candidates with a factor of 1/3. 9

10 6 FIGURES Figure 3: In this figure we show that the new function sin(π x 5 ) will reduce the twin probability with a factor Figure 4: In this figure we show the impact of prime 7 on the twin prime counting function. By imaginary shifting sin(π x 7 ) from values 7-14 to 14-21, we see that only in 2 out of the 7 cases, this shift would not have an impact on the possible twins. 10

11 6 FIGURES Figure 5: In this figure we show the results of our calculations and compare our estimate T win calc with the real T win Count for primes going to 999,999,

12 6 FIGURES Figure 6: In this figure we show the results of our calculations and compare our estimate T win calc with the real T win Count for primes going to 999,999,191 and we also included the calculation using π as a correction factor ( Twin Calc 2 ). 12

13 6 FIGURES Figure 7: In this figure we demonstrate how we have build up our excel of calculations. Table 1: In this table, we calculate the probability for a new twin prime immediately after a new prime ( Tot prob ) using formula (2), integrate it over the distance to the previous prime ( My pure ) and correct it with a factor π = to obtain My integr which is the approximation to Tw Count, the twin count value. P r Count T w Count P rime T ot prob My pure My integr

14 6 FIGURES Figure 8: In this figure we show the average distance between twin primes (in buckets of 100) for the first 403,905 twin primes upto twin 90,704,