The quadratic room and the divisors Mm

The study which, finding an infinite chain of suborders of natural numbers, all governed by particular divisors, solves Oppermann's conjecture (1882) and discovers the mathematical law that regulates the distribution of prime numbers.

Quadratic rooms and divisors Mm

The infinite succession of natural, integer and positive numbers forms a chain of numerical sub-orders, never detected by mathematicians, consisting of primitive mathematical structures prior to the decimal system used by man, made up of pairs of intervals having the same number of elements that systematically increase by one for each subsequent pair. The first pair of intervals, or "quadratic rooms", is made up of the numbers 1 (room A) and 2 (room B); the second from numbers 3, 4 (room A) and 5, 6 (room B); the third pair is made up of the numbers 7, 8, 9 (room A) and 10, 11, 12 (room B) and so on. All the elements of these pairs of intervals are governed by a particular divisor of all natural numbers called Mm (acronym for "Major of minors") and their analysis allows, among other things, to positively solve some of the biggest questions of the history of mathematics:

1)  Is there a mathematical law governing the distribution of prime numbers?

2) Why, as you move forward along the path that includes all natural numbers, the quantity of primes compared to the composed numbers is decreasing more and more?

3) There are linear mathematical structures that, apart from the demonstration provided by Euclid 2300 years ago, can make elementary and conceptually understandable even to first degree students the reason why prime numbers, even if constantly decreasing in relation to the numbers composed , are they meant to be infinite? 

Euclid, in the third century BC, managed to prove that the prime numbers are infinite. Two thousand years later, around 1800, the German mathematician Gauss invented a formula which, with excellent approximation, manages to quantify the prime numbers within any value of natural numbers. However, not having understood the mathematical law that determines the distribution of prime numbers, he delayed in revealing his discovery, fearing that from some point onwards the formula might begin to prove inaccurate. Fear remained unchanged and transmitted to the present day as no other mathematician had hitherto managed to discern from the natural succession of natural numbers that relationship governed by the divisors Mm operating within the sub-orders of the quadratic rooms, each of which always contains one or more prime numbers.

First key

The infinite list of natural numbers is littered with mathematical structures composed of pairs of intervals, that is, numbers that are consecutive to each other, defined as "quadratic rooms A and B", equipped with simple internal rules, never noticed by mathematicians because their primitive scaffold is earlier to the conventional decimal system adopted by man. By acquiring the concept of "quadratic rooms" you buy the first key; by studying the behavior of a particular divider common to the numerical elements that make up all the pairs of quadratic rooms, the second is acquired.

All numerical ranges called "quadratic rooms" are sub-orders of natural numbers. Each quadratic room A and B is made up of consecutive natural numbers that orbit each of the perfect squares, the first of which (quadratic room A) contains it as its last element, while the other (quadratic room B) consists of a quantity of elements, following the perfect square, equal to the previous one; this quantity of the elements that make up each quadratic room is always equal to the root of the perfect reference square; for example: the number of elements that make up the quadratic rooms A and B that refer to the perfect square 25, are 5 (number equal to the root of the perfect square), i.e. 21, 22, 23, 24, 25, for the quadratic room A, and 26, 27, 28, 29, 30, for the quadratic room B.

Graphical representation of the elements that make up the first 6 pairs A and B of the quadratic rooms:

                                         SQUARE ROOMS A - SQUARE ROOMS B

Second key

A particular type of divisor of natural numbers is identified and distinguished which is defined as Mm (acronym of "Major of minors") consisting of the major of the minors of all the numerical pairs that divide each natural number, bearing in mind the various possible cases:

a) All natural numbers are always divisible by one or more pairs of numbers. Composed numbers always have two or more pairs of divisors. For example, the pairs of divisors of the number 12 are three:

  1x12,

2x6,

3x4.

Taking care to always arrange the smaller divisor first in such pairs, it is easy to identify the largest of the minor ones, which, in this case, is the 3, therefore divisor Mm.

b) In the case of numerical elements corresponding to square numbers their divisor Mm always corresponds to its square root. For example, the pairs of divisors of the quadratic number 16 are three:

  1x16,

2x8,

4x4.

Among these three pairs it is easy to identify that the major divisor among the minor ones is 4, the square root of 16, therefore divisor Mm.

c) Prime numbers always have only one pair of divisors of which one is constituted by the number itself and the other constituted by the number 1. For example, the pair of divisors of the number 17 is

1x17,

therefore the divisor Mm is 1.

d) Some elements of the quadratic rooms sometimes have two or more divisors Mm because the cadenced pitch of the divisors greater than 1 sometimes flows into the same element, and this fact causes a duplication of presences of prime numbers within the same room; For example, the pairs of divisors of the number 18 are three:

  1x18,

2x9,

3x6.

Among these three pairs, it is easy to identify that the largest divisor among the minor ones is 3, which is certainly the divisor Mm of element 18; however, according to a mathematically explainable internal logic (which in the book containing the theory is described in detail) since all the divisors from 1 to n (where n always corresponds to the square root of the quadratic element) are present inside the quadratic room, it is evident that also the 2 is to be considered also divisor Mm (confluent) of the element 18.

It should be noted in this regard that although the number 2 is also divisor of the element 20, it does not take on the function of divider Mm since the divider Mm of 20 is a multiple thereof, i.e. the 4, being that in the presence of multiple divisors, between their submultiples, of the same dividing element, the function of divider Mm is generally assumed by the element of greater value. Therefore, being the element 20 divisible both by 4 and by 2, the function of divider Mm is assumed by 4. The confluence of two first divisors Mm (2, 3) in the same element (18) determines the repetition of the divisor Mm = 1 between the remaining elements of the same quadratic room (17, 19). Consequently in the interval [17, 20] there are two elements prime numbers (17, 19) instead of one.    

Quadratic rooms and divisors Mm

                                     A                          !                     B

The subscript numbers of the elements that make up the quadratic rooms indicate those particular divisors Mm of the elements belonging to the quadratic rooms. The elements that have the subscript number 1 are prime numbers.

The detailed analysis of the divisors Mm of the elements that make up all the quadratic rooms, for any value of n, allows us to detect that all the divisors from 1 to n are distributed on the elements of each quadratic room (where n always corresponds to the square root of the quadratic element that characterizes them) for which, for example, the divisors Mm of the five elements that make up both the quadratic room A and the quadratic room B of n = 5 are constituted by the numbers 1, 2, 3, 4, 5. In fact, for the quadratic room A, composed of the elements 21, 22, 23, 24, 25,

1 is the divisor Mm of the element 23,

2 is the divisor Mm of the element 22,

3 is the divisor Mm of the element 21,

4 is the divisor Mm of the element 24,

5 is the divisor Mm of the element 25,

while, as regards room B, composed of the elements 26, 27, 28, 29, 30,

1 is the divisor Mm of the element 29,

2 is the divisor Mm of the element 26,

3 is the divisor Mm of the element 27,

4 is the divisor Mm of the element 28,

5 is the divisor Mm of the element 30.

The same mathematical analysis makes it possible to verify that, for each value of n, small or large that it is, as a whole, the elements that make up the quadratic rooms together have a number of divisors Mm equal to the same value as n. For example: the elements of the quadratic room A of n = 100 consisting of the consecutive numbers that starting from 9901 reach 10,000 and the elements of the quadratic room B of the same n = 100 which starting from 10,001 arrive at 10,100, overall between their divisors Mm group all numbers between 1 and 100.

Since each value of n is always followed by a subsequent one and therefore the sequence of pairs of quadratic rooms is infinite, since the divisor Mm = 1 is systematically present in each quadratic room and since it is specific to the prime numbers, then the connection of the divisors Mm to the quadratic rooms, with the perennial presence of at least one element with the divisor 1, makes understandable the reason why the prime numbers are destined to be infinite.

The suborder of natural numbers, represented by their subdivision into quadratic rooms, becomes evident when, by extrapolating the relevant dividers Mm from their dividend elements that make up the quadratic rooms themselves, they are arranged in the same quadratic table, according to their natural cardinal order. In this way, a hidden suborder of natural numbers is highlighted, which recites a sing-song numeric monologue:

1, 1;  1, 2, 1, 2;  1, 2, 3, 1, 2, 3;

monologue that originally, during the spring of 2009, inspired by the discovery, I poetically defined "the heartbeat of the heart of numbers".

Table of divisors Mm of the first 56 natural numbers

Multiple subsequent discoveries made in the following years, which lasted until 2019 support, with demonstrations, the goodness of the original intuition described so far.

The mathematical rules, the same for each quadratic room, determined by the divisors Mm, attribute the presence of at least one prime number for each quadratic room A and B. In the first 7 there is only one prime number for each of them. Subsequently, as the number of elements composing them increases in the rooms, the presence of divisors Mm converging on some elements of the quadratic room determines a fluctuating increase in the presence of prime numbers for each room. The structures of the quadratic rooms which, at each passage from a value of n to its next, increase by a further numerical element with divisor Mm different from 1, make the reason for the rarefaction of the prime numbers conceptually clear.

On the other hand, even the empirical tests conducted on thousands of consecutive quadratic rooms, which abundantly confirm the assumption, are supported by the mathematical demonstration which, using two distinct and antithetical properties of the shape numbers 6k ± 1, manages to explain in detail both the certain presence of the prime numbers for each quadratic room and the univocal tendency towards rarefaction. To this must be added the mathematical formulas found that allow you to find, without fail, certain divisors Mm of different elements that make up the quadratic rooms, something is a sure indication of a further hidden suborder of natural numbers that supports the conception main.

On the other hand, the graph of the famous Spiral of Ulam also leads in this direction, which, unlike what was supposed by its Polish inventor who sought uninterrupted paths of prime numbers in it, correctly interpreted, instead becomes a generous source of indications that lead both at the numerical intervals called quadratic rooms and at the relevant divisors Mm.

Filippo Giordano -       The quadratic rooms and the divisors Mm.        Elementary theory of the sub-orders of natural numbers, mathematical law that regulates the distribution of prime numbers.