## Book

The Rainbow of Primes

Unfolding the Mystery of the Prime Numbers

Unfolding the Mystery of the Prime Numbers

**MAA Review**- Read more »

What are prime numbers? How are they related to physics? And why should we care about them? This concise though friendly book contains the answers to the previous questions and many others, explained from a unique point of view. Joao da Silva begins with a brief review of the history of mathematics, which inevitably introduces the well-known primes. He tell us that just as atoms are the basic building blocks of matter in our physical universe, prime numbers are their arithmetical counterpart in the world of mathematics. He then shows that such numbers are everywhere, from the moment we go online to make any sort of electronic transaction and back to the beginning of time itself.

The author analyzes the primes by taking an interdisciplinary approach combined with an original perspective, making it possible to delve deep into the intrinsic properties of these mysterious numbers. Such an unconventional method of scientific research applied to the so-called random distribution of the primes leads to astonishing correlations between these seemingly abstract numbers and various real-life disciplines such as experimental mathematics and computational number theory. Several chapters are dedicated to the implications of such findings, at both theoretical and practical levels. The author concludes with the discovery of his largest personal primes.

The book was published in December 2009 by Freund Publishing House Ltd. and was reviewed by the Mathematical Association of America (MAA) in 2010 as shown in the link below.

» mathdl.maa.org

» mathdl.maa.org

João Carlos Leandro da Silva obtained his Bachelor of Science degree in Electrical Engineering from the University of Connecticut (USA) and pursued graduate studies in theoretical physics at the Instituto Superior Técnico in Portugal. The author worked as a professional engineer in several countries and his passion for mathematics led him to present a poster abstract entitled "

**The distribution of prime numbers hidden within a fractal binary pattern**" at the Fractal 2004 conference in Vancouver, Canada. At present, he is a consultant in cryptography.### TABLE OF CONTENTS

Chapter 1:

Can Mathematicians be Poets?

Chapter 2:

How Can Both Time and Space be Connected to the Prime Numbers?

Chapter 3:

Helping to Secure Data Over the World Wide Web

Chapter 4:

Transforming Numbers into Primes

Chapter 5:

Fermat, Mersenne and Goldbach

Chapter 6:

Prelude to Breakthrough: An Interdisciplinary Approach

Chapter 7:

Finding the Rainbow of Primes

Chapter 8:

The Rebirth of Experimental Mathematics

Chapter 9:

Collatz or the 1-2-3 Problem?

Chapter 10:

The Factoring Problem and Some Algorithms

Chapter 11:

Proving that Large Primes are Indeed Prime Numbers

Chapter 12:

Largest Personal Primes

Table of First Four Thousand Primes

Answers to Questions in the Text

References

Index

Answers to Questions in the Text

**Appendix:**Chapters Containing Mathematica CodeReferences

Index

## 3D INTEGERS

Please note that in the above three-dimensional model the colors are different from those shown in the homepage of my website. Such is possible because the choice of colors is arbitrary. Due to our limited visual space, my model starts at zero and finishes at sixty-three but this unique representation of the positive integers continues until infinity. In other words, as the numbers get larger our model increases in both length and width but its maximum height remains constant. For example, the number twenty-nine is represented by the following vertical sequence of objects: an orange cube followed by an empty space and three consecutive pillars (dark green, brown and yellow), respectively. The brown pillar is composed of eight cubes and is the tallest while the orange cube is the shortest. In fact, the brown pillar represents the maximum height of the given three-dimensional model.

How were these 3D integers generated? Basically, only three steps are necessary to achieve such model. The first step consists in converting each positive integer into its binary representation and displaying the entire scheme not horizontally as it is often done but vertically. Next, substitute every binary zero with a white square and every binary one with a black square. The second step consists in counting the number of adjacent black squares surrounding every single black square. For example, the integer 3 converts to a vertical sequence of two binary ones which in turn become a column consisting of two black squares. Likewise, the integer 2 can be represented as a column of a black square and a white square and the integer 4 as a column of one black square followed by two white squares. Therefore, the most significant bit of 3 is the top black square which has three adjacent black squares (top digit from 2, bottom digit from 3 and top digit from 4). Similarly, the least significant bit of 3 is the bottom black square which has two adjacent black squares (the top digit from 3 and the top digit from 2).

As a result, the integer 3 is now transformed into a vertical sequence of decimal digits equal to the numerical value of thirty-two. The third and final step consists in substituting each digit of the previously mentioned sequence by a predefined color and height in order to obtain a three-dimensional object. For example, in this given model, the number one corresponds to the color dark blue and height one; the number two corresponds to the color violet and height two; the number three corresponds to the color green and height three; the number four corresponds to the color blue and height four; the number five corresponds to the color yellow and height five; the number six corresponds to the color pink and height six; the number seven corresponds to the color red and height seven and finally the number eight corresponds to the color brown and height eight. For example, the integer 63 is represented by the following vertical sequence of objects: six consecutive pillars (purple, blue, yellow, yellow, yellow and blue) as shown below in the bottom right-hand corner of the figure.

## 3D CRYPTOGRAPHY

### What is 3D cryptography ?

This is a new type of cryptography which enables the encryption and decryption of messages in three-dimensions. Let's go back in time and think about public-key cryptography. The typical scenario is two users (here called X and Y) where X wants to send a secret message to Y and vice-versa. For the moment, we do not care about authentication. Both users need to communicate over an insecure channel such as the Internet but the endpoints are assumed to be safe. Endpoints usually refer to the physical location of the devices that generate the messages to be transmitted. When public-key cryptography was invented, computers existed as devices that were physically located in a permanent position such as a mainframe in an office building. Since access to such facility was protected, the mainframe or endpoint was considered to be safe.Today, computers exist as mobile devices and, consequently, both users and devices are constantly changing their physical locations which brings about a completely different landscape. In fact, the present situation is the reverse of the previously described scenario since the endpoints are insecure but the communication channel is safe because we now know how to secure it. For example, user X is the owner of smartphone1, while user Y owns smartphone2. We assume that X is a male and Y is a female and both users are far away from each other. Consider user X alone inside his house with all doors and windows locked. If X picks smartphone1 from the kitchen table and writes a message, we can state that X is in a secure zone because nobody can see his message nor get hold of his phone. So what do we mean by insecure zones? Now consider user Y together with her friends outside a restaurant. If Y sends a message from smartphone2, we can easily state that Y pertains to an insecure zone because someone can see her message or take possession of her phone. It is obvious that both X and Y can freely move from a secure to an insecure zone and vice-versa. 3D cryptography assures that SMS privacy is guaranteed between any two mobile users.

## 3D INTEGER FACTORIZATION

**Factoring People**- Read more »

This is a new factorization technique specially designed for semiprimes with the final aim to factor composite numbers such as RSA-1024 bits in a reasonable time.

### A) PRIME SPACE

Prime space is the infinite space containing all primes starting at 2 and going to infinity. In our particular case, N = P x Q where N is a composite number, P and Q are primes and P > Q. As a result, we are interested in all the multiples of both P and Q.### B) FUNCTION SPACE

Function space is a new mathematical formula that relates three independent variables such that many potential candidates are automatically generated.### C) 3D SPACE

3D space represents the space where all potential candidates are found.### D) ALGORITHM

Algorithm is a highly optimized code that efficiently joins the previously mentioned spaces: prime, function and 3D.### E) SEARCH STATEGY

Since the algorithm can be of dual nature: deterministic or random and sequential or parallel, several different search strategies can be adopted.### F) PRIME FACTORS

The prime factor P or Q is obtained using an elementary mathematical operation.## THE CUBE OF GOD

The above cube has no faces. It has only edges and vertices.
Every vertex correspond to a specific physical event.
Every edge represents a transition from one event to another. Green-colored lines stand for transitions which obey the laws of physics. Red-colored lines symbolize transitions which break the laws of physics.
As a result, these lines are not edges but diagonals.
Arrows indicate the direction of time with respect to physical event.

The PDF below presents a unique argument for the origin of everything, answering a wide range of questions such as: Where is God? What is reality? Why are we here? What is the fate of the universe? What are we? What is life? What is death? How are miracles possible? What is mathematics? What is the limit of physics?

Since we have discovered a bridge between science and religion, it is now easy to decode the following sentences written by the great Portuguese poet, Fernando Pessoa (1888 - 1935):

"

At last, we realize that surprisingly both artists and scientists arrive at the same amazing conclusions. Yet, they use radically different approaches.

### Where does science meet religion?

At a point before the Big Bang called singularity. In my humble opinion, God is the singularity. So, what is God? Fundamentally, God consists of two parts: one physical and another transcendental. The physical part are all the known and unknown forces of physics, whereas the transcendental part is a higher conscious. Consequently, God is a unique and perfect entity. It has no gender and, as a whole, is beyond the comprehension of humans.The PDF below presents a unique argument for the origin of everything, answering a wide range of questions such as: Where is God? What is reality? Why are we here? What is the fate of the universe? What are we? What is life? What is death? How are miracles possible? What is mathematics? What is the limit of physics?

Since we have discovered a bridge between science and religion, it is now easy to decode the following sentences written by the great Portuguese poet, Fernando Pessoa (1888 - 1935):

"

**I am named the Absolute. Speak it not, figure it not; think and thou wilt find.**"At last, we realize that surprisingly both artists and scientists arrive at the same amazing conclusions. Yet, they use radically different approaches.

## NEWS

On January 26, 2014, I published on my website a PDF entitled "

On January 27-28, 2014, I attended the CYBERTECH 2014 (international conference & exhibition) which took place in Tel Aviv, Israel.

On February 24-25, 2014, I attended the International Security Conference (operational security tactics & solutions) which took place in Rome, Italy.

On June 16, 2014, I published on my website a PDF entitled "

**3D cryptography**" which is freely available for download.On January 27-28, 2014, I attended the CYBERTECH 2014 (international conference & exhibition) which took place in Tel Aviv, Israel.

On February 24-25, 2014, I attended the International Security Conference (operational security tactics & solutions) which took place in Rome, Italy.

On June 16, 2014, I published on my website a PDF entitled "

**The cube of God**" which presents a unique argument in favor of the existence of God and explains where science meets religion.## BIOGRAPHY

1966-1979

I was born on December 31 in Lisbon, Portugal. Since my father was a career diplomat, I lived in many different countries.

1980-1984

I attended the American International School of Lisbon.

1989

I obtained a bachelor degree in electrical engineering from the University of Connecticut in the USA.

1990-1991

I did my military service in Portugal.

1992-1995

I enrolled as a graduate student at the Instituto Superior Técnico in Lisbon. I completed all course work for the master degree in physics while working as a teaching assistant. I was for a short period a scientific associate at CERN.

1996-1998

I worked as an engineer for Samsung Corporation in Portugal.

1999-2003

I worked as an engineer for several smaller companies in Italy.

2004

I presented a poster abstract entitled, "

**The distribution of prime numbers hidden within a fractal binary pattern**" at the Fractal 2004 conference in Canada.2005

I attended the NKS summer school which took place at Brown University in the USA. My project concerned the Collatz problem.

2006-2008

I worked as an independent consultant in cryptography.

2009

My book was published by Freund Publishing House Ltd. and dedicated in memory of my father.

2010

I started my own company and attended the IEEE 26-th Convention of Electrical and Electronics Engineers in Israel where I presented two papers "

**Carmichael numbers and a new primality test**" and "**Factoring semiprimes and possible implications for RSA"**.2011-2013

I attended the Hacking Defined Experts (HDE 39) course in Israel and later invented a new technology called

**3D cryptography**based on my book.**NKS - Summer School -**Read more »

**IEEE - Advancing Technology for Humanity -**Read more »

**OEIS -**Read more »

## CONSULTING

I do consulting in the following areas:

ETHICAL HACKING/PENETRATION TESTING

CRYPTOGRAPHY

CRYPTANALYSIS

ALGORITHM DESIGN

MATHEMATICA PROGRAMMING

I have a multidisciplinary background in electrical engineering, computer science, theoretical physics and experimental mathematics. I worked in both industry and academia. Presently, my company does research in various fields with the objective to create new products and services. If you have any questions, please do not hesitate to contact me.

ETHICAL HACKING/PENETRATION TESTING

CRYPTOGRAPHY

CRYPTANALYSIS

ALGORITHM DESIGN

MATHEMATICA PROGRAMMING

I have a multidisciplinary background in electrical engineering, computer science, theoretical physics and experimental mathematics. I worked in both industry and academia. Presently, my company does research in various fields with the objective to create new products and services. If you have any questions, please do not hesitate to contact me.

## CONTACTS

If you wish to contact the author, please send an email to:

joaodasilva@rainbowofprimes.com

If you have a question regarding an order, please send an email to:

sales@rainbowofprimes.com

## TERMS OF USE

### Restrictions on Use

This Web site is owned and operated by Joao Carlos Leandro da Silva. By accessing or using this site, you agree to the terms set forth in this Terms of Use agreement as changed from time to time. Joao Carlos Leandro da Silva retains the right, at his discretion, to change any of these terms in the future.### Ownership of Copyright

This site contains material that is owned or licensed by Joao Carlos Leandro da Silva. You may not copy, reproduce, upload, post, distribute, republish, retransmit, or modify in any way any of the material on this site, including computer code and software.### Exclusion of Liability

YOUR USE OF THIS WEB SITE IS "AS IS", AND JOAO CARLOS LEANDRO DA SILVA MAKES NO REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED, ABOUT HIS SITE OR THE MATERIAL CONTAINED OR REFERRED TO ON IT, AND WILL NOT BE HELD LIABLE IN ANY WAY FOR YOUR USE OF IT. TO THE FULLEST EXTENT PERMISSIBLE PURSUANT TO APPLICABLE LAW, JOAO CARLOS LEANDRO DA SILVA DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. JOAO CARLOS LEANDRO DA SILVA FURTHER DOES NOT WARRANT THAT ITS WEB SITE OR THE PRODUCTS AND MATERIALS CONTAINED ON SUCH SITE ARE ERROR-FREE, OR THAT ERRORS WILL BE CORRECTED, OR THAT THIS SITE OR THE SERVER FROM WHICH IT IS RUN IS FREE OF VIRUSES OR OTHER POTENTIALLY HARMFUL CODES. UNDER NO CIRCUMSTANCES, INCLUDING NEGLIGENCE, SHALL JOAO CARLOS LEANDRO DA SILVA BE HELD LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL OR CONSEQUENTIAL DAMAGES AND EXPENSES WHATSOEVER (INCLUDING WITHOUT LIMITATION, HEALTH PROBLEMS, LOST PROFITS, AND DAMAGES ARISING FROM LOST DATA, COMPUTER VIRUSES, DELETION OF FILES OR BUSINESS INTERRUPTION) ARISING OUT OF OR IN ANY WAY CONNECTED WITH THE USE OF THIS SITE, OR FOR ANY INFORMATION, SOFTWARE, PRODUCTS OR SERVICES OBTAINED THROUGH THE RAINBOWOFPRIMES.COM SITE, WHETHER SUCH DAMAGES ARE BASED ON CONTRACT, TORT, STRICT LIABILITY OR OTHERWISE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. JOAO CARLOS LEANDRO DA SILVA RESERVES THE RIGHT TO CHANGE OR DISCONTINUE AT ANY TIME ANY ASPECT OR FEATURE OF THIS WEB SITE.### Links to Other Web Sites

Joao Carlos Leandro da Silva has provided links with other internet sites that are maintained by third parties, and may from time to time provide materials from such third party sites on this Web site. Joao Carlos Leandro da Silva does not in any way control the material contained on or the services or products offered through such third party sites. These terms and conditions apply solely to the Rainbow of Primes site and you should always review the terms and conditions of any site that you access through a link from this site.### Termination

This agreement may be terminated by Joao Carlos Leandro da Silva at any time without notice. In the event of termination, you are no longer authorized to have access to the site, and all limitations on Joao Carlos Leandro da Silva's liability and on your use of his material shall survive.## PRIVACY POLICY

### Summary

Joao Carlos Leandro da Silva takes your online privacy very seriously. You may visit this site anonymously as often as you like without providing any personally identifying information such as your name or email address. I do not collect personally identifying information from you except when you specifically provide it, and I protect all the information that I receive.### The Information You Provide

If you choose to purchase my book, I will need to collect additional information for billing and shipping, such as your full name, physical address and email address.### Links to Other Sites

My site contains links to other sites that are not under my control, even though they may refer to my person or to my book. Such other sites do not necessarily follow my privacy policies, and may place their own cookies on your computer. This practice is standard on the web, and I do not control cookies placed on your computer by third parties whose advertising or links appear on my site.### This website is maintained by

LEANDRO DA SILVA JOAO CARLOSVia Medole 22 - 46043 Castiglione delle Stiviere (MN) - Italy

I can be reached by email: webmaster@rainbowofprimes.com

Please, include the words "

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