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Prime numbers are the integers that cannot be divided exactly by another integer other than one and itself. Prime numbers are notoriously disobedient to rules: they seem to be randomly distributed among natural numbers with no laws except that of chance. Questions about prime numbers have been perplexing mathematicians over centuries. How to efficiently predict greater prime numbers has been a great challenge for many. Most of the previous studies focus on how many prime numbers there are in certain ranges or patterns of the first or last digits of prime numbers. Honestly, although these patterns are true, they help little with accurately predicting new prime numbers, as a deviation at any digit is enough to annihilate the primality of a number. The author demonstrates the periodicity and inter-relationship underlying all prime numbers that makes the occurrence of all prime numbers predictable. This knowledge helps to fish all prime numbers within one net and will help to speed up the related research.

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 [

Definition 1. The nth prime number is denoted as P_{n}. For example, P_{1} = 2,P_{4} = 7.

Definition 2. The product of the first n-1 prime numbers is designated as Super Product of P_{n}, denoted as X_{n}. For example, X_{1} = 1,X_{4} = 2 × 3 × 5 = 30.

Theorem 1. All prime numbers smaller than P_{n} cannot divide exactly the sum of an integer multiple of X_{n} and a prime number greater than or equal to P_{n}.

Proof. Suppose a prime number q ≥P_{n}, another prime number d <P_{n}, and an integer S = a × X_{n} + q (a ∈N).

Since d|X_{n}, then d|(a × X_{n}), and d ∤ q, therefore d ∤ (a × X_{n} + q), namely, d ∤ S. This completes the proof.

Theorem 2. All prime numbers smaller than P_{n} cannot divide exactly the sum of an integer multiple of X_{n} and the product of prime numbers greater than or equal to P_{n}.

The proof of this theorem is similar to that of Theorem 1.

Note. Both theorems imply 1) that a prime number smaller than P_{n} cannot divide exactly the sum specified in the theorems; 2) that, however, a prime number greater than or equal to P_{n} 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.

The above theorems suggest that X_{n} 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).

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 P_{n}. 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 (

It is interesting to extract the following generalizations.

1)Stage N starts with P_{n}.

2) Prime numbers deployed on the same radii of the circles have fixed difference of 2 (step length inStage N = X_{n}) 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

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.

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.

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).

The author declares no conflicts of interest regarding the publication of this paper.

Wang, X. (2021) The Genesis of Prime Numbers—Revealing the Underlying Periodicity of Prime Numbers. Advances in Pure Mathematics, 11, 12-18. https://doi.org/10.4236/apm.2021.111002