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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.

Primes appear to distribute randomly and they catch much attention from mathematicians for long time [

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 (

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} [

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.

Sums | Paired primes (#1 + #2) | #pairs |
---|---|---|

220 | 23 + 197, 29 + 191, 41 + 179, 47 + 173, 53 + 167, 71 + 149, 83 + 137, 89 + 131, 107 + 113 | 9 |

218 | 7 + 211, 19 + 199, 37 + 181, 61 + 157, 67 + 151, 79 + 139 | 6 |

216 | 5 + 211, 17 + 199, 19 + 197, 23 + 193, 37 + 179, 43 + 173, 53 + 163, 59 + 157, 67 + 149, 79 + 137, 89 + 127, 103 + 113, 107 + 109 | 13 |

214 | 3 + 211, 17 + 197, 23 + 191, 41 + 173, 47 + 167, 83 + 131, 101 + 113 | 7 |

212 | 13 + 199, 19 + 193, 31 + 181, 61 + 151, 73 + 139, 103 + 109 | 6 |

210 | 11 + 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 + 107 | 19 |

208 | 11 + 197, 17 + 191, 29 + 179, 41 + 167, 59 + 149, 71 + 137, 101 + 107 | 7 |

206 | 7 + 199, 13 + 193, 43 + 163, 67 + 139, 69 + 137, 79 + 127, 97 + 109 | 7 |

204 | 5 + 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 + 103 | 14 |

202 | 3 + 199, 5 + 197, 11 + 191, 23 + 179, 29 + 173, 53 + 149, 71 + 131, 89 + 113 | 8 |

200 | 3 + 197, 7 + 193, 19 + 181, 37 + 163, 43 + 157, 61 + 139, 73 + 127, 97 + 103 | 8 |

198 | 5 + 193, 7 + 191, 17 + 181, 19 + 179, 31 + 167, 41 + 157, 47 + 151, 59 + 139, 61 + 137, 67 + 131, 71 + 127, 89 + 109, 97 + 101 | 13 |

196 | 3 + 193, 5 + 191, 17 + 179, 23 + 173, 29 + 167, 47 + 149, 59 + 137, 83 + 113, 89 + 107 | 9 |

194 | 3 + 191, 13 + 181, 31 + 163, 37 + 157, 43 + 151, 67 + 127 | 6 |

192 | 11 + 181, 13 + 179, 19 + 173, 29 + 163, 41 + 151, 43 + 149, 53 + 139, 61 + 131, 79 + 113, 83 + 109, 89 + 103, | 11 |

190 | 11 + 179, 17 + 173, 23 + 167, 41 + 149, 53 + 137, 59 + 131, 83 + 107, 89 + 101 | 8 |

188 | 7 + 181, 31 + 157, 37 + 151, 61 + 127, 79 + 109 | 5 |

186 | 5 + 181, 7 + 179, 13 + 173, 19 + 167, 23 + 163, 29 + 157, 37 + 149, 47 + 139, 59 + 127, 73 + 113, 79 + 107, 83 + 103, 89 + 97 | 13 |

184 | 3 + 181, 5 + 179, 11 + 173, 17 + 167, 47 + 137, 53 + 131, 71 + 113, 83 + 101 | 8 |

182 | 3 + 179, 19 + 163, 31 + 151, 43 + 139, 73 + 109, 79 + 103 | 6 |

180 | 7 + 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 + 97 | 14 |

178 | 5 + 173, 11 + 167, 29 + 149, 41 + 137, 47 + 131, 71 + 107 | 6 |

176 | 3 + 173, 13 + 163, 19 + 157, 37 + 139, 67 + 109, 73 + 103, 79 + 97 | 7 |

174 | 7 + 167, 11 + 163, 17 + 157, 23 + 151, 37 + 137, 43 + 131, 47 + 127, 61 + 113, 67 + 107, 71 + 103, 73 + 101 | 11 |

172 | 5 + 167, 23 + 149, 41 + 131, 59 + 113, 71 + 101, 83 + 89 | 6 |

170 | 3 + 167, 7 + 163, 13 + 157, 19 + 151, 31 + 139, 43 + 127, 61 + 109, 67 + 103, 73 + 97 | 9 |

168 | 5 + 163, 11 + 157, 17 + 151, 19 + 149, 29 + 139, 31 + 137, 37 + 131, 41 + 127, 59 + 109, 61 + 107, 67 + 101, 71 + 97, 79 + 89 | 13 |

166 | 3 + 163, 17 + 149, 29 + 137, 53 + 113, 59 + 107 | 5 |

164 | 7 + 157, 13 + 151, 37 + 127, 61 + 103, 67 + 97 | 5 |

162 | 5 + 157, 11 + 151, 13 + 149, 23 + 139, 31 + 131, 53 + 109, 59 + 103, 61 + 101, 73 + 89, 79 + 83 | 10 |

160 | 3 + 157, 11 + 149, 23 + 137, 29 + 131, 47 + 113, 53 + 107, 59 + 101, 71 + 89 | 8 |

158 | 7 + 151, 19 + 139, 31 + 127, 61 + 97 | 4 |

156 | 5 + 151, 7 + 149, 17 + 139, 19 + 137, 29 + 127, 43 + 113, 47 + 109, 53 + 103, 59 + 97, 67 + 89, 73 + 83 | 11 |

154 | 3 + 151, 5 + 149, 17 + 137, 23 + 131, 41 + 113, 47 + 107, 53 + 101, 71 + 83 | 8 |

152 | 3 + 149, 13 + 139, 43 + 109, 73 + 79 | 4 |
---|---|---|

150 | 11 + 139, 13 + 137, 19 + 131, 23 + 127, 37 + 113, 41 + 109, 43 + 107, 47 + 103, 53 + 97, 61 + 89, 67 + 83, 71 + 79 | 12 |

148 | 11 + 137, 17 + 131, 41 + 107, 47 + 101, 59 + 89 | 5 |

146 | 7 + 139, 19 + 127, 37 + 109, 43 + 103, 67 + 79 | 5 |

144 | 5 + 139, 7 + 137, 13 + 131, 17 + 127, 31 + 113, 37 + 107, 41 + 103, 43 + 101, 47 + 97, 61 + 83, 71 + 73 | 11 |

142 | 3 + 139, 5 + 137, 11 + 131, 29 + 113, 41 + 101, 53 + 89, 59 + 83 | 7 |

140 | 3 + 137, 13 + 127, 31 + 109, 37 + 103, 43 + 97, 61 + 79, 67 + 73 | 7 |

138 | 7 + 131, 11 + 127, 29 + 109, 31 + 107, 37 + 101, 41 + 97, 59 + 79, 67 + 71 | 8 |

136 | 5 + 131, 23 + 113, 29 + 107, 47 + 89, 53 + 83 | 5 |

134 | 3 + 131, 7 + 127, 31 + 103, 37 + 97, 61 + 73 | 5 |

132 | 5 + 127, 19 + 113, 23 + 109, 29 + 103, 31 + 101, 43 + 89, 53 + 79, 59 + 73, 61 + 71 | 9 |

130 | 3 + 127, 17 + 113, 23 + 107, 29 + 101, 41 + 89, 47 + 83, 59 + 71 | 7 |

128 | 19 + 109, 31 + 97, 61 + 67 | 3 |

126 | 13 + 113, 17 + 109, 19 + 107, 23 + 103, 29 + 97, 37 + 89, 43 + 83, 47 + 79, 53 + 73, 59 + 67 | 10 |

124 | 11 + 113, 17 + 107, 23 + 101, 41 + 83, 53 + 71 | 5 |

122 | 13 + 109, 19 + 103, 43 + 79 | 3 |

120 | 7 + 113, 11 + 109, 13 + 107, 17 + 103, 19 + 101, 23 + 97, 31 + 89, 37 + 83, 41 + 79, 47 + 73, 53 + 67, 59 + 61 | 12 |

118 | 5 + 113, 11 + 107, 17 + 101, 29 + 89, 47 + 71 | 5 |

116 | 3 + 113, 7 + 109, 13 + 103, 19 + 97, 37 + 79, 43 + 73 | 6 |

114 | 5 + 109, 7 + 107, 11 + 103, 13 + 101, 17 + 97, 31 + 83, 41 + 73, 43 + 71, 47 + 67, 53 + 61 | 10 |

112 | 3 + 109, 5 + 107, 11 + 101, 23 + 89, 29 + 83, 41 + 71, 53 + 59 | 7 |

110 | 3 + 107, 7 + 103, 13 + 97, 31 + 79, 37 + 73, 43 + 67, 47 + 63 | 7 |

108 | 5 + 103, 7 + 101, 11 + 97, 19 + 89, 29 + 79, 37 + 71, 41 + 67, 47 + 61 | 8 |

106 | 3 + 103, 5 + 101, 17 + 89, 23 + 83, 47 + 59 | 5 |

104 | 3 + 101, 7 + 97, 31 + 73, 37 + 67, 43 + 61 | 5 |

102 | 5 + 97, 13 + 89, 19 + 83, 23 + 79, 29 + 73, 31 + 71, 41 + 61, 43 + 59 | 8 |

100 | 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53 | 6 |

98 | 19 + 79, 31 + 67, 37 + 61 | 3 |

96 | 7 + 89, 13 + 83, 17 + 79, 23 + 73, 29 + 67, 37 + 59, 43 + 53 | 7 |

94 | 5 + 89, 11 + 83, 23 + 71, 41 + 53 | 4 |

92 | 3 + 89, 13 + 79, 19 + 73, 31 + 61 | 4 |

90 | 7 + 83, 11 + 79, 17 + 73, 19 + 71, 23 + 67, 29 + 61, 31 + 59, 37 + 53, 43 + 47 | 9 |

88 | 5 + 83, 17 + 71, 29 + 59, 41 + 47 | 4 |

86 | 3 + 83, 7 + 79, 13 + 73, 19 + 67 | 4 |

84 | 5 + 79, 11 + 73, 13 + 71, 17 + 67, 23 + 61, 31 + 53, 37 + 47, 41 + 43 | 8 |

82 | 3 + 79, 11 + 71, 23 + 59, 29 + 53 | 4 |

80 | 7 + 73, 13 + 67, 19 + 61, 37 + 43 | 4 |

78 | 5 + 73, 7 + 71, 11 + 67, 17 + 61, 19 + 59, 31 + 47, 37 + 41 | 7 |
---|---|---|

76 | 3 + 73, 5 + 71, 17 + 59, 23 + 53, 29 + 47 | 5 |

74 | 3 + 71, 7 + 67, 13 + 61, 31 + 43 | 4 |

72 | 5 + 67, 11 + 61, 13 + 59, 19 + 53, 29 + 43, 31 + 41 | 6 |

70 | 3 + 67, 11 + 59, 17 + 53, 23 + 47, 29 + 41 | 5 |

68 | 7 + 61, 31 + 37 | 2 |

66 | 5 + 61, 7 + 59, 13 + 53, 19 + 47, 23 + 43, 29 + 37 | 6 |

64 | 3 + 61, 5 + 59, 11 + 53, 17 + 47, 23 + 41 | 5 |

62 | 3 + 59, 19 + 43 | 2 |

60 | 7 + 53, 13 + 47, 17 + 43, 19 + 41, 23 + 37, 29 + 31 | 6 |

58 | 5 + 53, 11 + 47, 17 + 41 | 3 |

56 | 3 + 53, 13 + 43, 19 + 37 | 3 |

54 | 7 + 47, 11 + 43, 13 + 41, 17 + 37, 23 + 31 | 5 |

52 | 5 + 47, 11 + 41, 23 + 29 | 3 |

50 | 3 + 47, 7 + 43, 13 + 37, 19 + 31 | 4 |

48 | 5 + 43, 7 + 41, 11 + 37, 17 + 31, 19 + 29 | 5 |

46 | 3 + 43, 5 + 41, 17 + 29 | 3 |

44 | 3 + 41, 7 + 37, 13 + 31 | 3 |

42 | 5 + 37, 11 + 31, 13 + 29, 19 + 23 | 4 |

40 | 3 + 37, 11 + 29, 17 + 23 | 3 |

38 | 7 + 31 | 1 |

36 | 5 + 31, 7 + 29, 13 + 23, 17 + 19 | 4 |

34 | 3 + 31, 5 + 29, 11 + 23 | 3 |

32 | 3 + 29, 13 + 19 | 2 |

30 | 7 + 23, 11 + 19, 13 + 17 | 3 |

28 | 5 + 23, 11 + 17 | 2 |

26 | 3 + 23, 7 + 19 | 2 |

24 | 5 + 19, 7 + 17, 11 + 13 | 3 |

22 | 3 + 19, 5 + 17 | 2 |

20 | 3 + 17, 7 + 13 | 2 |

18 | 5 + 13, 7 + 11 | 2 |

16 | 3 + 13, 5 + 11 | 2 |

14 | 3 + 11 | 1 |

12 | 5 + 7 | 1 |

10 | 3 + 7 | 1 |

8 | 3 + 5 | 1 |

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 [

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 ) (

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 _{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

Taking advantage of the above described pairing relationship between primes, routine generating greater primes includes the following steps (using

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