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.

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.

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.

The nth prime is denoted as P_n. The product of the first n – 1 primes is designated as Super Product of P_n, and denoted as X_n [9].

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

Proof.

1) By definition, X_n 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 (X_n – 1) and (X_n + 1) are primes, they constitute twin primes.

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 X_n – a and X_n + a are composites for all a ∈ { P i | 1 ≤ i < n } .

4) If a is a composite smaller than P_n, 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 X_n – a and X_n + a are composites.

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

This completes the proof.

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

In case none of X_n – 1 and X_n + 1 is a prime, (X_n – P_n, X_n + P_n) is a 2P_n long prime gap. An interesting inference is “As P_n 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 [15]. Considering the known greatest prime is more than 24 million digits long [6], 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 [4] [11].

Assuming X_n + 1 and X_n + P_n both are primes, let’s try to search for the next prime after X_n. Starting from X_n, the next number is X_n + 1 which is a prime. We finish the search with only one test, with 100% success rate. Starting from X_n + 2, we cannot succeed until reaching X_n + P_n. The success rate is 2/(P_n – 2) if we only test all the odds in the range. This rate decreases as the value of P_n 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 X_n – 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 X_n – a is a prime, as it may be divided exactly by a prime equal or greater than P_n. This is the shortcoming of this paper, namely, we cannot eliminate the all influence of numbers greater than P_n. However, it does imply that X_n – a is very likely a prime. This constitutes the rationality underlying the bilateral symmetrical distribution of primes shown in Figures 3-5.

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 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 [P_n, S/P_n]: if any of their products falls in (P_n, 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, P_n = 7, S/P_n = 4, the range [P_n, S/P_n] 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.

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 (X_n or its integer multiples).

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

Step 3. Calculate all combination products in (P_n, S) of all primes in [P_n, S/P_n]. 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.

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.

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

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.

The authors declare no conflicts of interest regarding the publication of 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