Symmetrical Distribution of Primes and Their Gaps

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1. Introduction

Primes appear to distribute randomly and they catch much attention from mathematicians for long time [1] - [11]. The Goldbach Conjecture states that every even number greater than 4 is a sum of two primes. A corollary of the conjecture is that every single integer greater than 3 is equally distanced from two primes, implying that at least two primes are paired on both sides of an integer. This pattern has been documented in previous publications [10] [12] [13]. However, how many prime pairs there are on both sides of an integer remains to be an unanswered question. Trying to answer this question, we found that more primes tend to be paired on both sides of the halves of super products (or their integer multiples) of primes. Many mathematicians have noted the existence of prime gaps as well as twin primes (which have a difference of 2 in between) [5] [14], but whether there is any regularity about the occurrence of such gaps remains an open question. Here we demonstrate that the pairwise (bilateral symmetrical) distributions of primes and their gaps near the super products (or its integer multiples), hoping it will trigger more interesting investigations.

2. Methods

Initially, the target of our investigation was to figure out how many prime pairs have the same sum. The statistics indicated pair number peaks at super products of primes (or their integer multiples). We analyzed and proved the rationality underlying these peaks, and proved the existence of gaps around super products of primes (or their integer multiples) and validated a routine generating primes.

3. Results

After we manually obtained number of prime pairs for every even number under 220 (Table 1) and did statistics (Figure 1), it became obvious that there are local peaks at 30, 60, 90, 120 …, namely, the number peaks periodically.

4. Theoretical Analysis and Proof

The *n*th prime is denoted as *P _{n}*. The product of the first

Theorem 1. There are at most two primes, namely, *X _{n}* – 1 and

Proof.

1) By definition, *X _{n}* is a composite.

2) Since
$a|{X}_{n}$ and
$a\nmid 1$ for all
$a\in \left\{{P}_{i}|1\le i<n\right\}$, therefore
$a\nmid \left({X}_{n}-1\right)$ and
$a\nmid \left({X}_{n}+1\right)$. As
$\left({X}_{n}+1\right)=\left({X}_{n}-1\right)+2$, so if both (*X _{n}* – 1) and (

Figure 1. Number of prime pairs for each even number under 220 demonstrates a general increasing trend, and it peaks periodically at the multiples of 30. Note that all the numbers are equal to or greater than 1, the number required by the Goldbach Conjecture.

Table 1. The first 107 even numbers and the paired primes.

3) Since
$a|{X}_{n}$ and
$a|a$ for all
$a\in \left\{{P}_{i}|1\le i<n\right\}$, therefore
$a|\left({X}_{n}-a\right)$ and
$a|\left({X}_{n}+a\right)$, namely, all *X _{n}* –

4) If *a* is a composite smaller than *P _{n}*,

In summary, *X _{n}* – 1 and

This completes the proof.

Note. *X _{n}* – 1 and

In case none of *X _{n}* – 1 and

Assuming *X _{n}* + 1 and

Theorem 2. For
$a\in \left\{{P}_{i}|i\ge n\right\}$ and
$b\in \left\{{P}_{i}|1\le i<n\right\}$, *b* cannot divide *X _{n}* –

Proof.

Since $b|{X}_{n}$ and $b\nmid a$, therefore $b\nmid \left({X}_{n}-a\right)$.

Figure 2. Prime gap (*X _{n}* –

This completes the proof.

Note. This does not necessarily mean that *X _{n}* –

5. Implications on Relationship between Primes

As implied by Theorem 2, primes can be paired under certain condition. Now we designate *S* as a constant equal to *X _{n}* (or its integer multiples), then

Note 1. Although 1 is not a prime, its complementary may be a prime.

Note 2. Some of the numbers in (*S*/2, *S*) obtained by subtracting smaller primes may be composites. These exceptions can be eliminated case-by-case by calculating all combination products of all primes in [*P _{n}*,

Note 3. As shown in Figure 3, when *S* = 30, *P _{n}* = 7,

6. Algorithm Generating Primes

Taking advantage of the above described pairing relationship between primes, routine generating greater primes includes the following steps (using Figure 3 as an example).

Figure 3. Prime pairs with sums equal to 30, with no exception.

Figure 4. Prime pairs with sums equal to 60, with one exception of 49 (=7 × 7).

Figure 5. Prime pairs with sums equal to 210, with four exceptions of 121 (=11 × 11), 143 (=11 × 13), 169 (=13 × 13), and 187 (=11 × 17).

Step 1. Calculate the value of *S* (*X _{n}* or its integer multiples).

Step 2. Subtract 1 and all primes in [*P _{n}*,

Step 3. Calculate all combination products in (*P _{n}*,

Step 4. Save the above generated list. Finish.

7. Discussions

Compared to the existing routines generating primes, the present one has the following advantages:

1) The calculation involved is computationally cheap. The candidate list of greater primes can be obtained by subtracting smaller ones from a constant.

2) The result is dense, namely, all primes within scope are covered.

3) Although the applicable range of each run is limited, the applicable range of the routine can be extended exponentially into the infinite, as it is hinged with super products of primes.

8. Conclusion

Primes tend to be pairwise distributed. Such pairing relationship implies that greater primes can be obtained in a computationally cheap way. There is either one continuous 2*P _{n}* long prime gap or two at least

Acknowledgements

This research was supported by the Strategic Priority Research Program (B) of Chinese Academy of Sciences (Grant No. XDB26000000), and National Natural Science Foundation of China (41688103, 91514302). We appreciate the constructive suggestions from two anonymous reviewers and Mr. Wuwei Wang.

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