The following information related to the use of two simple formulas and trigonometric functions will allow us to learn the principle of creating prime numbers and predict their error-free occurrence bypassing any actions related to the priority test. In short, we'll learn the long-sought prime numbers code.

The most optimal formulas for determining two number sequences in which over 73% of natural numbers are omitted and which will determine all prime numbers and "alongside" prime numbers are:

n ≠ 0

and

n ≠ 0

Numbers from the set 6n+1 :

7, 13, 19 , 31, 37, 43, 49, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 169, 181, 187, 193, 199, 211, 217, 223,...

Numbers from the set 6n-1 :

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 77, 83, 89, 101, 107, 113, 119, 131, 137, 143, 149, 161, 167, 173, 179, 191, 197, 203, 209, 221,...

As we can see in the picture below, we place our two sets with prime numbers and "alongside" prime numbers on the X coordinate axis. From the intersection of the X and Y axes to the left we place the numbers from the formula 6n-1, while on the right we insert the numbers from the formula 6n+1 .

Now a properly modified trigonometric function, e.g. sine, comes to our aid. We determine from our smallest number, that is, from the number "5" a sine wave that crosses the X axis every 5 positions both to the right and to the left. In the picture below we see the effect of this action:

As we can see above, the places where the sine wave intersects the X axis are our prime numbers which are not prime numbers because they are squares of prime numbers and / or "prime numbers" or their products. As we can see, the next number greater than "5" is the number "7" and we can certainly say before determining the next sine wave that all natural numbers on our X axis smaller than the value of the square of the number "7", which is less than "49" through which the sine wave passes are prime numbers. The next step will be to determine the sine wave that will intersect the X axis every seven positions from the number "7".

We see again that all natural numbers through which none of the sinusoids determined so far pass - smaller than the value of "121" (that is, the square of our next number which is the number "11") are certainly the next prime numbers.

Using this method indefinitely, i.e. from the number "5" determining the sinusoide which intersects the X axis every five positions, from the number "7" sinusoide intersecting the X axis every seven positions, from the number "11" the sinusoid crossing the X axis every eleven positions, etc. we determine without error in advance prime numbers smaller than the square of the next number from which we finished determining the sine wave. All natural numbers on our X axis through which at least two sine waves have passed are not prime numbers, and all those through which the sine wave has run only once are prime numbers. Remember to determine sinusoide gradually, i.e. first from "5", later "7", "11", "13", "...". and do not omit under any circumstances the numbers through which another sine wave has already passed, for example the numbers "25", "35", "..."

The free Graph program for graphs of functions can be downloaded here and a ready file with the functions described above here.

You can use a method that contains all natural numbers to determine successive primes where the principle that after determining the sine from a specific number will work, all numbers smaller than the square of the next nutural number through which the sine does not run are prime numbers, but this method forces us for operations on all natural numbers as shown in the picture below;

The free Graph program for graphs of functions can be downloaded here and a ready file with the functions described above here.

The code of prime numbers and how they arise was known (broken) thanks to trigonometric functions that are much more efficient for computers to calculate than complex mathematical calculations. In addition, the use of formulas 6n-1 and 6n+1 reduces the search for prime numbers to 33% of all natural numbers.

All number squares from the set "6n+1" and "6n-1" always appear only in the set 6n+1. In addition, the product of any two numbers from the set 6n+1 is always found in this set.

The set "6n-1" is more "pure" because apart from prime numbers it contains only products of prime numbers where one factor is always the number from the set 6n-1 and the second factor is the number from the set 6n+1.

The product of any two numbers from the set 6n-1 always appears in the set 6n+1.

Each number that occurs in the set 6n-1 to check if it is a prime number is enough to divide only by the numbers from the set 6n+1, which speeds up checking whether the number is a prime number.