The quadratic rooms. Elementary theory of the distribution of prime numbers - Filippo Giordano - ebook

Versione inglese contenente una parte dei capitoli dell’opera “Le stanze quadratiche”. Il più antico enigma sui numeri naturali riguarda il mistero della formazione dei numeri primi. Questo studio ne risolve il mistero con l'introspezione dei numeri naturali, mettendo in risalto un particolare tipo di loro divisori, definiti Mm, i quali sono caratteristica costante degli intervalli numerici orbitanti attorno ai numeri quadrati. Tali intervalli, che costituiscono i vari "Insiemi Ima e Imb", facilmente identificabili nella scala dei numeri naturali, sono sempre più grandi e contengono sempre più numeri primi. Corredato da molteplici tabelle esplicative, lo studio evidenzia anche delle formule matematiche che consentono di individuare, mediante i loro divisori, le postazioni dei numeri composti all’interno di tali intervalli. Inoltre mette in evidenza la straordinaria concomitanza di elementi degli insiemi IMA e IMB coi raggruppamenti numerici che si formano in ciascun lato degli infiniti quadrati che si ricavano dalla “Spirale” del matematico polacco Stanislaw Ulam.

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Theoretical synthesis

The divisors Mm

The divisors Mm (table)

Multiple Sets A (MS-A) and Multiple Sets B (MS-B)

Relationship between Set A, Set B. Set T

Confluent divisors and replicating divisors

Detail of the MA and MB sets of the first 24 values

Some common features of MS-A and MS-B


Summary of quadratic rooms and Mm divisors element, 1 to 441

The layout of the dividers Mm of the sets Ms-A and MS-B

MS-A divider form

Note to MS-A divider form

MS-B divider form

Note to MS-B divider form

Table of first MS-A values with higher divisors

Table of first MS-B values with higher divisors

Stanislaw Ulam Spiral, SM-A and SM-B

Stanislaw Ulam’s Spiral

Number of prime numbers within MS-A and MS-B sets

The secret heartbeat of the natural numbers

Contrary to what we think, the prime numbers do not come out randomly, but they are under a mathematical law that places the prime numbers within precise numerical intervals that I, about eight years ago, have identified, calling them "quadratic rooms".

From then on, I continued the study and made several other collateral discoveries, all consistent with the theory, including the Stanislaw Ulam Spiral, if properly observed.

Therefore, this book is a precious collection of new and true verifiable information on the first numbers that make it clear why they are intended to be infinite and why the estimation of the number of prime numbers made by Gauss will be uniform even for numerical regions that are currently unknown to us.

However, knowing the mathematical law that governs the distribution of the prime numbers, contrary to what is commonly thought, does not endanger computer security since the prime numbers are the last of which one can locate their placement within the Ima and Imb sets. The larger the number of elements that make up the sets and the longer the time it takes to sort them through the Mm divisors. In practice, you know in what container you have to look for them, but in order to find them you must first identify all of their many roommates, all of which are mathematically identifiable, and only in the end, by exclusion, you get to them. This confirms that it is impossible to find them through a fast math formula that can allow them to reach them directly. The knowledge of the mathematical law that regulates the distribution of the prime numbers therefore does not endanger the security of the computer, which the banks entrust all the economic transactions taking place in the world, but it makes sure the impossibility of decrypting rapidly first used numbers.

Filippo Giordano


elementary theory of the distribution of prime numbers


At some point in his story, for convenience of calculation, man invented the decimal system. The decimal system method, which entered the mental form of human civilization, however, did not allow to decipher the riddle of the rule of distribution of the prime numbers within natural numbers.

God, or, if you like, Nature, has thought about distributing numbers in a different form. Attention: My words do not refer to mathematical rules or even the natural succession of numbers. I just say that to understand the mathematical mechanism that governs the distribution of the prime numbers we have to think of a different format than the decimal one.

With the decimal system invented by man, the subtle appearance of the prime numbers in the context of natural numbers appears unconnected with any rule. There is no math formula that can predict where they are. There is no way for them to decide how and why their occasional presence sometimes happens to be more compact. From number 90 to number 100 we find only one prime number. From number 100 to number 110 there are four prime numbers. A great puzzle for over two thousand years, which keeps mathematicians from around the world in distress.

In 1751, the great Euler expressed this: “There are some mysteries that the human mind will never penetrate. To be convinced, we must only look at the tables of the prime numbers. We will find that there is no reign nor order nor law”

After the Euclid theorem, which in 300 BC he succeeded in demonstrating that the prime numbers are infinite, the greatest progress made on this field of mathematics was achieved by the German Gauss, who around 1800 could calculate, with an approximate formula, in what proportion the prime numbers are present to all the other numbers. Then, in spite of the fact that in the last 70 years, large computing faculty of the world's mathematics faculties are looking for ever larger numbers, no significant progress has been made to decipher the mystery self that surrounds the birth of the prime numbers.

But since the decimal system used in our civilization has prevented mathematicians from deciphering the secret of the prime numbers, what is the other system that allows us to come to the fore of the most excellent problem in the world of mathematics? The quadratic system !

What is the quadratic system? If you only know this definition today ... be quiet! It is not your particular cultural gap as it is the result of my original interpretation of the distribution of the prime numbers.

A special way to arrange the numbers not yet encoded by the mathematical sciences I have seen through a primitive and elementary path of mathematics. Primitive path through which you can finally see a great light that illuminates infinite quadratic rooms where ever and forever live all the prime numbers. This is, therefore, a primitive path that leads to the place where, after dislodging the last branch of the tree born with the invention of the decimal system (which prevents the view) appears, as Nature created it, the world of the prime numbers in all its splendor.

Of this I speak in this my basic theory of natural numbers homogeneously distributed in quadratic rooms that always host the prime numbers. Interpretable numerical distribution through the identification of their gametic codes, that is, those precise pairs of factors that form the same numbers.

A phenomenon that, properly framed, is always faithful. This is a valid phenomenon for all structurally identified consecutive quadratic number sets, which allow you to take full consciousness of the mathematical dynamics that gives rise to the prime numbers.

This phenomenon I identified in the spring of 2009, and later, I was more and more convinced of its powerful light, until, in 2013, I casually discovered it found a great ally in an unknown application of the Spiral by Stanislaw Ulam.

Theoretical synthesis

Imagine that the sequence of natural numbers, which starts from number 1 to infinity, is arranged in pairs of homogeneous numerical sets. Imagine that the constant characteristics of such pairs of homogeneous sets are, for the first set, those that comprise elements ranging from (n-1) n + 1 to n^2, including the extremes, and, secondly, to comprehending the elements ranging from n^2 + 1 to N(n + 1).

Note that, so arranged, each of the pairs of numerical sets identified refers to a different and progressive value of n, and that each of the sets is composed of a number of elements always equal to the corresponding value of n.

01 – 02

03, 04 – 05, 06

07, 08, 09 – 10, 11, 12

13, 14, 15, 16 – 17, 18, 19, 20

21, 22, 23, 24, 25 – 26, 27, 28, 29, 30

Each line consists of two numerical sets of which we call A the first Together, which in its interior includes the square number, and B the next one. The number of elements in such sets is always equal to the value of n.

After this premise we make a second premise.

All integers and natural numbers are produced by a pair of factors. The prime numbers are always the product of a single pair of factors, that is, the number 1 and the number itself. For example, the number 7, which is the prime number, is the product of 1 x 7; so the 7 is produced by 1 which multiplies the same 7.

Composite numbers are produced by two or more pairs of numbers.

For example:

number 10, which is a composite number, is the product of the pair of 1 x 10 and also of the pair of 2 x 5. The number 9, which is also composed and is the square of 3, is the product of the 1x 9 pair and of the 3 x 3 torque (that is a number multiplying itself); therefore, one of the pairs of factors from which each number is produced, if it is a square number, is made up of two equal numbers.

Numerical elements located within these ranges therefore have the obvious characteristic of having pairs of factors equal to n for the quadratic element, while all other elements have pairs of factors, one of which is less than n and the other is superior. For example, element 22, in addition to being the product of the 1x22 pair, being composed has another pair of factors, and in fact is the product of 2 x 11.

213 – 222 – 231 – 244 – 255

262 – 273 – 284 – 291 – 305

For each set A and B, a sequence of factors below N (selected according to the criterion explained in the next paragraph) is always formed reflecting another T-set composed of a number of elements equal to that of the A and B sets ordered chronologically from 1 to N (1, 2, 3, ... n).