The Genesis of Prime Numbers—Revealing the Underlying Periodicity of Prime Numbers ()
1. Introduction
Prime numbers appear random and irregularly distributed, especially when viewed from a linear perspective. Many mathematicians have been studying the density of prime numbers such as function π(x), which has intrigued numerous publications [1], and focus on how many prime numbers there are in certain ranges. However, even if this function is well understood, it remains elusive as to exactly which one among numerous numbers is a prime number. There are many publications about the pattern of the first or last digit of prime numbers [2] [3]. These studies lack the power of prediction as the unique feature, primality, of prime numbers requires no error at any digit. Twin prime numbers are of interest for many, and mathematicians are eager to learn more about their distribution [4] [5] [6]. Although the currently known greatest prime number is already 24,862,048 digit-long [7], greater one remains a further target for many [1]. To help solve these problems, it is necessary to find patterns underlying prime numbers. Here I tried to demonstrate a novel periodic pattern underlying all prime numbers, and hoped to fuel the related research.
2. Promising Primality
Definition 1. The nth prime number is denoted as Pn. For example, P1 = 2,P4 = 7.
Definition 2. The product of the first n-1 prime numbers is designated as Super Product of Pn, denoted as Xn. For example, X1 = 1,X4 = 2 × 3 × 5 = 30.
Theorem 1. All prime numbers smaller than Pn cannot divide exactly the sum of an integer multiple of Xn and a prime number greater than or equal to Pn.
Proof. Suppose a prime number q ≥Pn, another prime number d <Pn, and an integer S = a × Xn + q (a ∈N).
Since d|Xn, then d|(a × Xn), and d ∤ q, therefore d ∤ (a × Xn + q), namely, d ∤ S. This completes the proof.
Theorem 2. All prime numbers smaller than Pn cannot divide exactly the sum of an integer multiple of Xn and the product of prime numbers greater than or equal to Pn.
The proof of this theorem is similar to that of Theorem 1.
Note. Both theorems imply 1) that a prime number smaller than Pn cannot divide exactly the sum specified in the theorems; 2) that, however, a prime number greater than or equal to Pn may divide exactly the sum specified in the theorems. The latter implication is the reason for exceptions in Figures 3-5. It is also the shortcoming of this paper, namely, I cannot eliminate all composite number exceptions a priori.
3. The Underlying Periodicity
The above theorems suggest that Xn is the step length between neighboring prime numbers on the same radius (in the same series) in Stage N, suggestive of periodicity underlying prime numbers. Following I will demonstrate the truthful existence of such periodicity of prime numbers stage by stage (Figures 1-5).
4. Inferences
More further stages are not allowed to be shown given the limited space of this paper. It is enough to say that more stages of prime numbers following similar regularity are there to show. Each stage takes all elements (except the first one) of the preceding stage as its initial ancestor circle (innermost). Every next outer derived circle (descendant circle) is generated by adding the step length of present stage to the elements of the preceding circle. This process is repeated multiple times until the total number of circles reaches Pn. Therefore, although the periodicity in a stage is restricted to its own stage, as there is no upper limit for the number of stages that is hinged with number of prime numbers (which is infinite, as proven), apparently, similar (although slightly different) regularity can be extended to the infinite (Figure 6).
It is interesting to extract the following generalizations.
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Figure 1. Stage I: 1 series starting with 2, step length 1 (the unit in integers), including 2 elements (2, 3).
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Figure 2. Stage II: 1 series, starting with 3, step length 2, including 3 elements (3, 5, 7).
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Figure 3. Stage III: 2 series (starting with 5 and 7, respectively), step length 6, including 10 elements (5, 11, 17, 23, 29; 7, 13, 19, 25, 31), with 1 twin prime number series and 1 exception (25).
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Figure 4. Stage IV: 8 series (starting with 7, 11, 13, 17, 19, 23, 29, 31, respectively), step length 30, including 56 elements (7, 37, 67, 97, 127, 157, 187; 11, 41, 71, 101, 131, 161, 191; 13, 43, 73, 103, 133, 163, 193; 17, 47, 77, 107, 137, 167, 197; 19, 49, 79, 109, 139, 169, 199; 23, 53, 83, 113, 143, 173, 203; 29, 59, 89, 119, 149, 179, 209; 31, 61, 91, 121, 151, 181, 211), with 3 twin prime number series (starting with 11 and 13, 17 and 19, 29 and 31, respectively) and 12 exceptions (49, 77, 91, 119, 121, 133, 143, 161, 169, 187, 203, 209).
1)Stage N starts with Pn.
2) Prime numbers deployed on the same radii of the circles have fixed difference (step length inStage N = Xn) with the neighboring ones.
3) All prime numbers are included within the generated series.
4) A prime number greater than 7 can be a sum of a composite number and a prime/composite number, except twin prime numbers with a difference 2 in
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Figure 5. Stage V: 48 series starting with 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179,181, 187, 191, 193, 197, 199, 209, 211, respectively, step length 210, including 528 elements, with 15 twin prime number series (starting with 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, 71 and 73, 101 and 103, 107 and 109, 137 and 139, 149 and 151, 167 and 169, 179 and 181, 191 and 193, 197 and 199, 209 and 211, respectively) and 188 exceptions.
between.
5) The existence of certain composite numbers exceptional to the regularity are necessary for the increasingly sparse density of prime numbers in later generations and for the genesis of some prime numbers.
6) Twin prime numbers give birth to new twin prime numbers.
7) It is intriguing to test whether there is a comparability between the genesis of prime numbers and that of other entities, such as organisms.
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Figure 6. The number of elements (A) and maximal prime number (B) in the stages increase exponentially with the number of stages.
5. Conclusion
If the numbers on the innermost circle in each stage are taken as ancestor prime numbers that give birth to more prime numbers, the numbers on the outer circles can be taken as descendants of these ancestors. The differences between neighboring descendants are fixed within a stage. The periodicity underlying prime numbers elaborated here makes prime numbers tame and predictable.
Acknowledgements
I thank the reviewers for their kind and constructive suggestions, which have improved this paper greatly. This research is supported by the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (XDB26000000) and the National Natural Science Foundation of China (41688103, 91514302).