APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2021.115031APM-109276ArticlesPhysics&Mathematics Symmetrical Distribution of Primes and Their Gaps BrandonY. Wang1XinWang2*School of Mechanical, Electrical &amp; Information Engineering, Shandong University, Weihai, ChinaState Key Laboratory of Palaeobiology and Stratigraphy, Nanjing Institute of Geology and Palaeontology and Center for Excellence in Life and Paleoenvironment, Chinese Academy of Sciences, Nanjing, China17052021110544745629, March 202122, May 2021 25, May 2021© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution-NonCommercial International License (CC BY-NC).http://creativecommons.org/licenses/by-nc/4.0/

Primes are of great importance and interest in mathematics partially due to their hard-to-predict distribution. A corollary of the Goldbach Conjecture is that two primes are equally distanced from a mid-point integer. Here the authors demonstrate that most primes are bilateral symmetrically distributed on the both sides of the halves of super products (or their integer multiples) of primes. This pattern suggests that greater primes may be obtained more efficiently by subtracting smaller ones from constants equal to super products (or their integer multiples) of primes.

Prime Number Distribution Bilateral Symmetry Super Product Pairwise
1. Introduction

Primes appear to distribute randomly and they catch much attention from mathematicians for long time  - . 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   . 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)  , 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 nth prime is denoted as Pn. The product of the first n – 1 primes is designated as Super Product of Pn, and denoted as Xn .

Theorem 1. There are at most two primes, namely, Xn – 1 and Xn + 1, in (Xn – Pn, Xn + Pn), and these two, if both valid, constitute twin primes.

Proof.

1) By definition, Xn is a composite.

2) Since a | X n and a ∤ 1 for all a ∈ { P i | 1 ≤ i < n } , therefore a ∤ ( X n − 1 ) and a ∤ ( X n + 1 ) . As ( X n + 1 ) = ( X n − 1 ) + 2 , so if both (Xn – 1) and (Xn + 1) are primes, they constitute twin primes.

The first 107 even numbers and the paired primes
SumsPaired primes (#1 + #2)#pairs
22023 + 197, 29 + 191, 41 + 179, 47 + 173, 53 + 167, 71 + 149, 83 + 137, 89 + 131, 107 + 1139
2187 + 211, 19 + 199, 37 + 181, 61 + 157, 67 + 151, 79 + 1396
2165 + 211, 17 + 199, 19 + 197, 23 + 193, 37 + 179, 43 + 173, 53 + 163, 59 + 157, 67 + 149, 79 + 137, 89 + 127, 103 + 113, 107 + 10913
2143 + 211, 17 + 197, 23 + 191, 41 + 173, 47 + 167, 83 + 131, 101 + 1137
21213 + 199, 19 + 193, 31 + 181, 61 + 151, 73 + 139, 103 + 1096
21011 + 199, 13 + 197, 17 + 193, 19 + 191, 29 + 181, 31 + 179, 37 + 173, 43 + 167, 47 + 163, 53 + 157, 59 + 151, 61 + 149, 71 + 139, 73 + 137, 79 + 131, 83 + 127, 97 + 113, 101 + 109, 103 + 10719
20811 + 197, 17 + 191, 29 + 179, 41 + 167, 59 + 149, 71 + 137, 101 + 1077
2067 + 199, 13 + 193, 43 + 163, 67 + 139, 69 + 137, 79 + 127, 97 + 1097
2045 + 199, 7 + 197, 11 + 193, 13 + 191, 23 + 181, 31 + 173, 37 + 167, 41 + 163, 47 + 157, 53 + 151, 67 + 137, 73 + 131, 97 + 107, 101 + 10314
2023 + 199, 5 + 197, 11 + 191, 23 + 179, 29 + 173, 53 + 149, 71 + 131, 89 + 1138
2003 + 197, 7 + 193, 19 + 181, 37 + 163, 43 + 157, 61 + 139, 73 + 127, 97 + 1038
1985 + 193, 7 + 191, 17 + 181, 19 + 179, 31 + 167, 41 + 157, 47 + 151, 59 + 139, 61 + 137, 67 + 131, 71 + 127, 89 + 109, 97 + 10113
1963 + 193, 5 + 191, 17 + 179, 23 + 173, 29 + 167, 47 + 149, 59 + 137, 83 + 113, 89 + 1079
1943 + 191, 13 + 181, 31 + 163, 37 + 157, 43 + 151, 67 + 1276
19211 + 181, 13 + 179, 19 + 173, 29 + 163, 41 + 151, 43 + 149, 53 + 139, 61 + 131, 79 + 113, 83 + 109, 89 + 103,11
19011 + 179, 17 + 173, 23 + 167, 41 + 149, 53 + 137, 59 + 131, 83 + 107, 89 + 1018
1887 + 181, 31 + 157, 37 + 151, 61 + 127, 79 + 1095
1865 + 181, 7 + 179, 13 + 173, 19 + 167, 23 + 163, 29 + 157, 37 + 149, 47 + 139, 59 + 127, 73 + 113, 79 + 107, 83 + 103, 89 + 9713
1843 + 181, 5 + 179, 11 + 173, 17 + 167, 47 + 137, 53 + 131, 71 + 113, 83 + 1018
1823 + 179, 19 + 163, 31 + 151, 43 + 139, 73 + 109, 79 + 1036
1807 + 173, 13 + 167, 17 + 163, 23 + 157, 29 + 151, 31 + 149, 41 + 139, 43 + 137, 53 + 127, 67 + 113, 71 + 109, 73 + 107, 79 + 101, 83 + 9714
1785 + 173, 11 + 167, 29 + 149, 41 + 137, 47 + 131, 71 + 1076
1763 + 173, 13 + 163, 19 + 157, 37 + 139, 67 + 109, 73 + 103, 79 + 977
1747 + 167, 11 + 163, 17 + 157, 23 + 151, 37 + 137, 43 + 131, 47 + 127, 61 + 113, 67 + 107, 71 + 103, 73 + 10111
1725 + 167, 23 + 149, 41 + 131, 59 + 113, 71 + 101, 83 + 896
1703 + 167, 7 + 163, 13 + 157, 19 + 151, 31 + 139, 43 + 127, 61 + 109, 67 + 103, 73 + 979
1685 + 163, 11 + 157, 17 + 151, 19 + 149, 29 + 139, 31 + 137, 37 + 131, 41 + 127, 59 + 109, 61 + 107, 67 + 101, 71 + 97, 79 + 8913
1663 + 163, 17 + 149, 29 + 137, 53 + 113, 59 + 1075
1647 + 157, 13 + 151, 37 + 127, 61 + 103, 67 + 975
1625 + 157, 11 + 151, 13 + 149, 23 + 139, 31 + 131, 53 + 109, 59 + 103, 61 + 101, 73 + 89, 79 + 8310
1603 + 157, 11 + 149, 23 + 137, 29 + 131, 47 + 113, 53 + 107, 59 + 101, 71 + 898
1587 + 151, 19 + 139, 31 + 127, 61 + 974
1565 + 151, 7 + 149, 17 + 139, 19 + 137, 29 + 127, 43 + 113, 47 + 109, 53 + 103, 59 + 97, 67 + 89, 73 + 8311
1543 + 151, 5 + 149, 17 + 137, 23 + 131, 41 + 113, 47 + 107, 53 + 101, 71 + 838
1523 + 149, 13 + 139, 43 + 109, 73 + 794
15011 + 139, 13 + 137, 19 + 131, 23 + 127, 37 + 113, 41 + 109, 43 + 107, 47 + 103, 53 + 97, 61 + 89, 67 + 83, 71 + 7912
14811 + 137, 17 + 131, 41 + 107, 47 + 101, 59 + 895
1467 + 139, 19 + 127, 37 + 109, 43 + 103, 67 + 795
1445 + 139, 7 + 137, 13 + 131, 17 + 127, 31 + 113, 37 + 107, 41 + 103, 43 + 101, 47 + 97, 61 + 83, 71 + 7311
1423 + 139, 5 + 137, 11 + 131, 29 + 113, 41 + 101, 53 + 89, 59 + 837
1403 + 137, 13 + 127, 31 + 109, 37 + 103, 43 + 97, 61 + 79, 67 + 737
1387 + 131, 11 + 127, 29 + 109, 31 + 107, 37 + 101, 41 + 97, 59 + 79, 67 + 718
1365 + 131, 23 + 113, 29 + 107, 47 + 89, 53 + 835
1343 + 131, 7 + 127, 31 + 103, 37 + 97, 61 + 735
1325 + 127, 19 + 113, 23 + 109, 29 + 103, 31 + 101, 43 + 89, 53 + 79, 59 + 73, 61 + 719
1303 + 127, 17 + 113, 23 + 107, 29 + 101, 41 + 89, 47 + 83, 59 + 717
12819 + 109, 31 + 97, 61 + 673
12613 + 113, 17 + 109, 19 + 107, 23 + 103, 29 + 97, 37 + 89, 43 + 83, 47 + 79, 53 + 73, 59 + 6710
12411 + 113, 17 + 107, 23 + 101, 41 + 83, 53 + 715
12213 + 109, 19 + 103, 43 + 793
1207 + 113, 11 + 109, 13 + 107, 17 + 103, 19 + 101, 23 + 97, 31 + 89, 37 + 83, 41 + 79, 47 + 73, 53 + 67, 59 + 6112
1185 + 113, 11 + 107, 17 + 101, 29 + 89, 47 + 715
1163 + 113, 7 + 109, 13 + 103, 19 + 97, 37 + 79, 43 + 736
1145 + 109, 7 + 107, 11 + 103, 13 + 101, 17 + 97, 31 + 83, 41 + 73, 43 + 71, 47 + 67, 53 + 6110
1123 + 109, 5 + 107, 11 + 101, 23 + 89, 29 + 83, 41 + 71, 53 + 597
1103 + 107, 7 + 103, 13 + 97, 31 + 79, 37 + 73, 43 + 67, 47 + 637
1085 + 103, 7 + 101, 11 + 97, 19 + 89, 29 + 79, 37 + 71, 41 + 67, 47 + 618
1063 + 103, 5 + 101, 17 + 89, 23 + 83, 47 + 595
1043 + 101, 7 + 97, 31 + 73, 37 + 67, 43 + 615
1025 + 97, 13 + 89, 19 + 83, 23 + 79, 29 + 73, 31 + 71, 41 + 61, 43 + 598
1003 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 536
9819 + 79, 31 + 67, 37 + 613
967 + 89, 13 + 83, 17 + 79, 23 + 73, 29 + 67, 37 + 59, 43 + 537
945 + 89, 11 + 83, 23 + 71, 41 + 534
923 + 89, 13 + 79, 19 + 73, 31 + 614
907 + 83, 11 + 79, 17 + 73, 19 + 71, 23 + 67, 29 + 61, 31 + 59, 37 + 53, 43 + 479
885 + 83, 17 + 71, 29 + 59, 41 + 474
863 + 83, 7 + 79, 13 + 73, 19 + 674
845 + 79, 11 + 73, 13 + 71, 17 + 67, 23 + 61, 31 + 53, 37 + 47, 41 + 438
823 + 79, 11 + 71, 23 + 59, 29 + 534
807 + 73, 13 + 67, 19 + 61, 37 + 434
785 + 73, 7 + 71, 11 + 67, 17 + 61, 19 + 59, 31 + 47, 37 + 417
763 + 73, 5 + 71, 17 + 59, 23 + 53, 29 + 475
743 + 71, 7 + 67, 13 + 61, 31 + 434
725 + 67, 11 + 61, 13 + 59, 19 + 53, 29 + 43, 31 + 416
703 + 67, 11 + 59, 17 + 53, 23 + 47, 29 + 415
687 + 61, 31 + 372
665 + 61, 7 + 59, 13 + 53, 19 + 47, 23 + 43, 29 + 376
643 + 61, 5 + 59, 11 + 53, 17 + 47, 23 + 415
623 + 59, 19 + 432
607 + 53, 13 + 47, 17 + 43, 19 + 41, 23 + 37, 29 + 316
585 + 53, 11 + 47, 17 + 413
563 + 53, 13 + 43, 19 + 373
547 + 47, 11 + 43, 13 + 41, 17 + 37, 23 + 315
525 + 47, 11 + 41, 23 + 293
503 + 47, 7 + 43, 13 + 37, 19 + 314
485 + 43, 7 + 41, 11 + 37, 17 + 31, 19 + 295
463 + 43, 5 + 41, 17 + 293
443 + 41, 7 + 37, 13 + 313
425 + 37, 11 + 31, 13 + 29, 19 + 234
403 + 37, 11 + 29, 17 + 233
387 + 311
365 + 31, 7 + 29, 13 + 23, 17 + 194
343 + 31, 5 + 29, 11 + 233
323 + 29, 13 + 192
307 + 23, 11 + 19, 13 + 173
285 + 23, 11 + 172
263 + 23, 7 + 192
245 + 19, 7 + 17, 11 + 133
223 + 19, 5 + 172
203 + 17, 7 + 132
185 + 13, 7 + 112
163 + 13, 5 + 112
143 + 111
125 + 71
103 + 71
83 + 51

3) Since a | X n and a | a for all a ∈ { P i | 1 ≤ i < n } , therefore a | ( X n − a ) and a | ( X n + a ) , namely, all Xn – a and Xn + a are composites for all a ∈ { P i | 1 ≤ i < n } .

4) If a is a composite smaller than Pn, a must have all of its prime factors b ∈ { P i | 1 ≤ i < n } . Since b | X n and b | a , therefore b | ( X n − a ) and b | ( X n + a ) , therefore all Xn – a and Xn + a are composites.

In summary, Xn – 1 and Xn + 1 are the only numbers in (Xn – Pn, Xn + Pn) that can be primes and with a difference of 2 in between.

This completes the proof.

Note. Xn – 1 and Xn + 1 does not necessarily be a prime, either of them may be divided exactly by a prime equal to or greater than Pn.

In case none of Xn – 1 and Xn + 1 is a prime, (Xn – Pn, Xn + Pn) is a 2Pn long prime gap. An interesting inference is “As Pn approaches to the infinite, length of the gap also approaches the infinite”. This answers the Question vii asked by Dr. Hua on page 90 of his book . Considering the known greatest prime is more than 24 million digits long , it is amazing to conceive that there exists such a long gap of primes: at most only two primes are immersed in zillions and zillions of composites! This provides one solution for the problem of prime gap (A8) and raw material for hypotheses on difference between consecutive primes mentioned in  .

Assuming Xn + 1 and Xn + Pn both are primes, let’s try to search for the next prime after Xn. Starting from Xn, the next number is Xn + 1 which is a prime. We finish the search with only one test, with 100% success rate. Starting from Xn + 2, we cannot succeed until reaching Xn + Pn. The success rate is 2/(Pn – 2) if we only test all the odds in the range. This rate decreases as the value of Pn grows greater. Knowing the above knowledge about prime gap, the test can be restricted to all odds in [ X n + P n − 1 + 1 , X n + P n ] , the success rate is 2 / ( P n − P n − 1 ) (Figure 2).

Theorem 2. For a ∈ { P i | i ≥ n } and b ∈ { P i | 1 ≤ i < n } , b cannot divide Xn – a exactly.

Proof.

Since b | X n and b ∤ a , therefore b ∤ ( X n − a ) .

This completes the proof.

Note. This does not necessarily mean that Xn – a is a prime, as it may be divided exactly by a prime equal or greater than Pn. This is the shortcoming of this paper, namely, we cannot eliminate the all influence of numbers greater than Pn. However, it does imply that Xn – a is very likely a prime. This constitutes the rationality underlying the bilateral symmetrical distribution of primes shown in Figures 3-5.

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 Xn (or its integer multiples), then S/2 can be a mid-point integer between of prime pairs on its both sides. The relationship between such prime pairs is termed complementary here since the sums of such pairs are always equal to a constant S, as shown in Figures 3-5.

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 [Pn, S/Pn]: if any of their products falls in (Pn, S/2), add the complementary of the product into the list; if any of their products falls in (S, S/2), delete the product from the generated list.

Note 3. As shown in Figure 3, when S = 30, Pn = 7, S/Pn = 4, the range [Pn, S/Pn] becomes [7,4], which is an empty range. This explains the lack of exceptions in Figure 3 although such exceptions occur in Figure 4 and Figure 5.

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).

Step 1. Calculate the value of S (Xn or its integer multiples).

Step 2. Subtract 1 and all primes in [Pn, S/2) from S, save the result in an ascending order as a list.

Step 3. Calculate all combination products in (Pn, S) of all primes in [Pn, S/Pn]. If such a product is within the list obtained in Step 2, delete it from the list.

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 2Pn long prime gap or two at least Pn – 1 long prime gaps around Xn. One or two of Xn – 1 and Xn + 1 may be the only primes within (Xn – Pn, Xn + Pn).

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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Wang, B.Y. and Wang, X. (2021) Symmetrical Distribution of Primes and Their Gaps. Advances in Pure Mathematics, 11, 447-456. https://doi.org/10.4236/apm.2021.115031

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