Squared prime numbers
Authors: Appelqvist Gunnar 
Published: February 2021

My investigation shows that there is a regularity even by the prime numbers. This structure is obvious when a prime square is created. The squared prime numbers.

1. Connections in a prime square

A prime square (or origin square) is defined as a square consisting of as many boxes as the origin prim squared. This prime settle every side of the square. So, for example, the origin square 17 have got four sides with 17 boxes along every side. The prime numbers in each of the 289 boxes are filled with primes when a prime number occur in the number series (1,2,3,4,5,6,7,8,9 and so on) and then is noted in that very box.

If a box is occupied in the origin square A this prime number could be transferred to the corresponding box in a second square B, and thereafter the counting and noting continue in the first square A. Eventually we get two filled prime squares. Analyzing these squares, you leave out the right vertical line, representing only the origin prime number,

When a square is filled with primes you subdivide it into four corner squares, as big as possible, denoted a, b, c and d clockwise. You also get a center line between the left and right vertical sides.

Irrespective of what kind of constellation you activate this is what you find:

Every constellation in the corner square a and/or d added to a corresponding constellation in the corner square b and/or c is evenly divisible with the origin prime.

Every constellation in the corner square a and/or b added to a corresponding constellation in the corner square d and/or c is not evenly divisible with the origin prime.

Every reflecting constellation inside two of the opposed diagonal corner squares, possibly summarized with any optional reflecting constellation inside the two other diagonal corner squares, is evenly divisible with the origin prime squared. You may even add a reflection inside the center line and get this result.

My Conjecture 1 is that this applies to every prime square without end.

 

A formula giving all prime numbers endless

 

In the second prime square the prime numbers are always higher than in the first square if you compare a specific box. There is a mathematic connection between the prime numbers in the first and second square. This connection appears when you square and double the origin prime and thereafter add this number to the prime you investigate. A new higher prime is found after n additions.

You start with lowest applicable prime number 3 and its square 3². Double it and you get 18. We add 18 to the six next prime numbers 5, 7, 11, 13, 17 and 19 in any order. After a few adds you get a prime and after another few adds you get another higher one. In this way you continue as long as you want to. The primes are creating themselves.

A formula giving all prime numbers is:

  5+18×n, +18×n, +18×n … without end

  7+18×n, +18×n, +18×n … without end

11+18×n, +18×n, +18×n … without end

13+18×n, +18×n, +18×n … without end

17+18×n, +18×n, +18×n … without end

19+18×n, +18×n, +18×n … without end

The letter n in the formula stands for how many 18-adds you must do until the next prime is found.

My Conjecture 2 is that this you find every prime number by adding 18 to the primes 5, 7, 11, 13, 17 and 19 one by one endless.

A method giving all prime numbers endless

 There is still a possibility to even more precise all prime numbers. You start a 5-number series derived from the start primes 7, 17, 19, 11, 13 and 5 in that very order. Factorized these number always begin with number 5. When each of these numbers are divided with five the quotient is either a prime number or a composite number containing of two or some more prime numbers in the nearby. By sorting out all the composite quotients you get all the prime numbers endless and in order.

Every composite quotient starts with a prime from 5 and up, squared. Thereafter the quotients starting with that prime show up periodically according to a pattern of short and long sequences. The position for each new prime beginning the composite quotient is this prime squared and multiplied with 5. Thereafter the short sequence is this prime multiplied with 10, while the long sequence is this prime multiplied with 20.

When all the composite quotients are deleted there are left several 5-numbers which divided with 5 give all prime numbers, and you even see clearly the distance between the prime numbers which for instance explain why the prime twins occur as they do.

My Conjecture 3 is that this is an exact method giving all prime numbers endless and in order.