the twin prime counting function

The twin prime counting function and it’s consequences

Chris De Corte

chrisdecorte@yahoo.com

February 23, 2015

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CONTENTS CONTENTS

Contents

1 Key-Words 3

2 Derivation of the formula 3

3 Calculation 4

4 Consequences 4

4.1 About the number of twin primes . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.2 Maximum distance between twin primes . . . . . . . . . . . . . . . . . . . . . 4

5 References 4

6 Figures 5

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2 DERIVATION OF THE FORMULA

1 Key-Words

Prime numbers, number theory, twin prime, conjecture, zahlentheorie, Brun, Hardy-Littlewood,

Polignac, Yitang Zhang; [1] .

2 Derivation of the formula

Suppose an empty x-axis with all the natural numbers still vacant. We can ask ourselves

what the chance would be that twin primes could be created. We can assume that this will

be 100% or 1.

Therefore, we introduce the first prime on our x-axis by adding a sinus wave with a period of

2 (figure 1). If we cross out all numbers where this wave goes through 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 = 1/2.

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 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 = 0.16667.

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 0.16667 · 3/5 = 0.1.

The reader can test the next step using figure 4.

We continue the same logic for the other prime numbers and derive a general formula:

probi = probi−1.pi − 2

pi(1)

In the previous we were still calculating individual probabilities. To know the total probability

of the occurrence of a twin prime, after a set of primes already in place, we integrate each of

the above formula over the length that they are valid:

totprobx =∑∀pi≤x

probi.(pi − pi−1) (2)

After testing the above formula for the first 1 million primes (above 86,027 twin primes or

until prime 15,485,863), we came to the conclusion that we have to correct our results with a

factor 3.1416. We call this correction factor π.

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5 REFERENCES

We now declare our twin prime counting formula and give it the new symbol Π2:

Π2(x ≥ 3) = π.∑∀pi≤x

(

pi∏3

probi−1.pi − 2

pi).(pi − pi−1) with prob2 = 0.5 (3)

3 Calculation

The calculation has been done in Microsoft Excel. An explanation can be found in figure 6.

The result can be seen in figure 5.

As an extra, we have calculated 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 7.

4 Consequences

4.1 About the number of twin primes

In figure 6, 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

the above implies the twin prime conjecture.

4.2 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,

the average distance will keep on increasing. This finding seems to be confirmed by the

general trend in figure 7 and by the trend line formula. So, the maximum distance between

twin primes will be unbounded.

5 References

1. en.wikipedia.org ; Twin Prime.

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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.

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6 FIGURES

Figure 3: In this figure we show that the new function sin(π x5 ) will reduce the twin probability with

a factor 5−25 .

Figure 4: In this figure we show the impact of prime 7 on the twin prime counting function. By

imaginary shifting sin(π x7 ) 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.

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6 FIGURES

Figure 5: In this figure we show the results of our calculations and compare our estimate twin est

with the real twin count for primes going to 15,485,857.

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6 FIGURES

Figure 6: In this figure we demonstrate how we have build up our excel of calculations.

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6 FIGURES

Figure 7: In this figure we show the average distance between twin primes (in buckets of 100) for the

first 403,905 twin primes upto twin 90,704,657.

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