The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. By using parametric sine functions, I will show here that all assumptions known today regarding the distribution of prime numbers larger than 3 are wrong. In particular, this refutes Riemann´s hypothesis of random distribution of prime numbers. Furthermore, I will show an exact primality test based on these parametric sine functions, which only needs the calculation modes +, -, · , : and only uses one parameter. There is no such deterministic primality test existing until today[3] [4].
Research Article
On the order of the non-primes of type 6n+5 and 6n+1
https://doi.org/10.21203/rs.3.rs-443272/v1
This work is licensed under a CC BY 4.0 License
Version 1
posted 21 Apr, 2021
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The number sequences 6n+5 (for all n
0) and 6n+1 (for all n
0) cover all odd natural numbers that are not divisible by 3. For example, 6n+5 leads to 5, 11, 17, ... and 6n+1 leads to 7, 13, 19, ... . Because all primes larger than 3 are odd natural numbers and not divisible by 3, they must be of type 6n+5 or 6n+1. All potential prime numbers larger than 3 can therefore be shown as zeros of the following two sine functions: f1(x) = sin ((
for all numbers of type 6n+5 and f2(x) = sin ((
for all numbers of type 6n+1. The allowed range for x reaches for both sine functions from 0 to infinite.
Now the question arises, which natural numbers of type 6n+5 and 6n+1 are not prime?
- Under what conditions are numbers of type 6n+5 not prime and what regularity do these follow?
6n+5 is not a prime number if: (6n+5) : a = b ↔ a · b = 6n+5 ↔ 
= n a,b,n 
N and a,b
A natural number is divisible by 6 if it is divisible by 2 and by 3. For the divisibility by 2, a and b must be odd natural numbers because an odd natural number multiplied by another odd natural number leads to an odd natural number. And an odd natural number larger than or equal to 7 minus 5 leads to an even natural number. For the divisibility by 3, it is necessary that a · b has the structure (3x+1)(3y-1) for all x,y N because
= 
= 
= 3xy-x+y – 2
Therefore, for a nonprime number of type 6n+5, the following must hold for the index n:
for all n 
and x, y even natural numbers 
For x = 2 and y = 2, 4, 6, 8, 10, ... after inserting n in 6n+5 this leads to the numbers 35, 77, 119, 161, 203, ... as you can see this is every 42nd number starting at 35.
For x = 4 and y = 2, 4, 6, 8, 10, ... after inserting n in 6n+5 this leads to the numbers 65, 143, 221, 299, 377, ... as you can see this is every 78th number starting at 65. Note, that 65 equals 35+30.
For x = 6 and y = 2, 4, 6, 8, 10, ... after inserting n in 6n+5 this leads to the numbers 95, 209, 323, 437, 551, ... as you can see this is every 114th number starting at 95. Note, that 95 equals 35+2 times 30.
This pattern continues indefinitely and can be represented by (35 + 30k) + (42 + 36k)m with k = 0, 1, 2, 3, ... and m = 0, 1, 2, 3, ...
All nonprimes of type 6n+5 can thus be represented as zeros of the following parametric sine function:
f3(x) = sin((
)
The allowed range for x in the parametric sine function f3(x) ranges from 35+30t to infinite, and for t, t = 0, 1, 2, 3, 4, ...
If x and y are interchanged, the same numbers are produced, but in a different order. Therefore, the interchange of the variables x and y can be ignored in this context. Nevertheless, I would like to point out that swapping x and y also leads to regular columns of numbers, which can be represented as zeros of the parametric sine function: f4(x) = sin((
)(x – 35 – 42t)). The allowed range for x reaches in f4(x) from 35+42t to infinite and for t applies t = 0, 1, 2, 3, 4, ...
- Under what conditions are numbers of type 6n+1 not prime and what regularity do these follow? 6n+1 is not a prime number if: (6n+1) : a = b ↔ a · b = 6n+1 ↔
= n a,b,n N and a,b1
Following the proof idea from above, a and b must be odd natural numbers to ensure divisibility by 2. For the divisibility by 3, it is necessary that a times b has either the form (3x+1)(3y+1) or the form (3x-1)(3y-1) because 
= 
= 
= 3xy+x+y and
= 
= 
= 3xy-x-y for all x,y N
Therefore, for a nonprime number of type 6n+1, the following must hold: either 
or 
for all n 
and x, y even natural numbers .
By using the values for x and y the same way as above, the first of those two terms produces the following columns of numbers:
49, 91, 133, 175, 217, ... then 91, 169, 247, 325, 403, ... then 133, 247, 361, 475, 589, ...
This pattern continues indefinitely and can be represented by (49 + 42k) + (42 + 36k)m with k = 0, 1, 2, 3, ... and m = 0, 1, 2, 3, ... .
This first part of the nonprime numbers of type 6n+1 can therefore be represented as zeros of the following parametric sine function:
f5(x) = sin((
The allowed range for x in the parametric sine function f5(x) ranges from 49+42t to infinite, and for t, t = 0, 1, 2, 3, 4, ...
The second term produces the following columns of numbers:
25, 55, 85, 115, 145, ... then 55, 121, 187, 253, 319, ... then 85, 187, 289, 391, 493, ...
Also this pattern continues indefinitely. It can be represented by (25 + 30k) + (30 + 36k)m with k = 0, 1, 2, 3, ... and m = 0, 1, 2, 3, ... .
This second part of nonprime numbers of type 6n+1 can therefore be represented as zeros of the following parametric sine function:
f6(x) = sin((
The allowed range for x in the parametric sinus function f6(x) ranges from 25+30t to infinite, and for t, t = 0, 1, 2, 3, 4, ...
To make all prime numbers appear in the correct order, only the functions f1(x), f2(x), f3(x), f5(x) and f6(x) have to be entered into a common coordinate system. The following figures are made with MatheGrafix 11. The zeros of the characteristic green functions show all potential prime numbers of type 6n+5 or type 6n+1. Every green zero (f1(x) and f2(x)), which is not crossed by the graph of one of the “forbidding” parametric sine functions (f3(x), f5(x), f6(x)), represents a prime number. The software used does not allow a parameter-dependent definition set. Therefore, these two figures have the same rigid definition range and because of these different parametric settings. Increasing the parameters stepwise is necessary when using this software because the “forbidding” parametric sinus functions logically cross out wrong numbers when the functions are under their definition range.
- A consequential possibility to prove whether a natural number larger than 3 is prime or not
- Proof of whether a natural number A under investigation is of type 6n+5 or of type 6n+1.
If (A-5) : 6 equals a natural number, then A is of type 6n+5. If (A-1) : 6 equals a natural number, then A is of type 6n+1. If both calculations do not equal a natural number, then the number under investigation is not of type 6n+5 or of type 6n+1. Such a natural number larger than 3 cannot be prime.
- a) If the investigated number A is of type 6n+5, calculate if f3(x) = sin((
)
has a zero at this number A. Then, A is not a prime number. To do this, it is enough to prove if A – 35 – 30t = 0 can be solved with a natural number value of t (including t=0) or if 
can be solved with a natural number result when using a natural number value of t (including t=0). If the first term shows a natural number for the value of t, f3(x) shows zero at 0; if the second term shows a natural number result for a natural value of t, f3(x) shows zero at an integer multiple of 
.
b) If the investigated number A is of type 6n+1, calculate if f5(x) = sin((
or f6(x) = sin((
has a zero at this number A. If one of these two functions shows a zero at this number A, then A is not prime.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Competing interests: The author declares no competing interests.
[1] Gauß, C. F. (1801). Disquisitiones Arithmeticae. Chapter 2. Retrieved from: Disquisitiones.ocr.pdf (uni-bielefeld.de).
[2] Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Größe. Monatsberichte der Berliner Akademie, November 1859. Transcripted by D. R. Wilkins, Preliminary Version: December 1998. Retrieved from: zeta.pdf (claymath.org)
[3] Steuding, J, Weng, A. (2004). Primzahltests. Von Eratosthenes bis heute. Retrieved from: (PDF) Primzahltests - von Eratosthenes bis heute (researchgate.net)
[4] Mcgregor-Dorsey, Zachary, S. (1999). Methods of Primality Testing. Retrieved from: (PDF) Methods of Primality Testing (researchgate.net)