Fermat’s Inception

The Fermat Project is part of a great story that started at ancient Greece with Pythagoras and of course it also involves Pierre de Fermat. Here I tell the complete story up to the very moment I became part of it.

Greek origins

Pythagoras had to pay his first student to convince him to study with him. He gave him three oboli (about a penny in today’s money) per lesson until one day he said there was no more money. The student replied that he preferred to pay, provided that they continued with the lessons. The rumor ran through the Greek islands and so the Pythagorean School, also known as the Secret Brotherhood, a philosophical/religious monastic order counting 300 members, was formed. By then counting and calculating had been widely practiced by the Babylonians and Egyptians but the Pythagoreans went far beyond that. “All is Number” was their motto attesting to their conviction that numbers were the key to life itself. Pythagoras strongly believed that relationships between numbers could reveal, through logical demonstration, all the secrets of the universe, hence he started producing his theorems. The Brotherhood walked briskly with his theorems but did nothing to share the secrets of the universe arousing the ire of the people who one day set the school on fire.

Despite the desperate attempt of his students to save him, Pythagoras himself died in the fire. The Brotherhood was dispersed until Alexander the Great founded Alexandria and in order to attract scholars to the new city he took the advice of his general, Ptolemy: “Gather the great books; great minds will follow.”

Hence the Library of Alexandria was established with the overall aim of collecting every book ever written. Each traveler had their books confiscated upon arrival to Alexandria. They would be handed over to scribes, who made a copy for the owner and sent the original to the library. The extensive collection comprising 500,000 volumes made Alexandria the intellectual capital of the ancient world and as Ptolemy had envisaged attracted the most famous scholars of the era.

Euclid was put in charge of the math section and by inventing the “reductio ad absurdum” that is the proof by contradiction, he took the findings of Pythagoras one step further. Anything that would defy logic seemed abominable to Pythagoras and as such was dismissed.

Though the Pythagoreans had discovered irrational numbers (pi, or the square root of two) such was their fear of the “unutterable” numbers that their study was immediately forbidden and the disciple who came up with the concept of the “square root of two” was executed. Euclid announced to everyone that Pythagoras was abominable and advocated that irrational numbers would open a new door for mathematics and as such encouraged them to think about them without fear.

The First Peer to Peer Network

Julius Caesar’s attack on Alexandria marked the demise of the Royal Library as it was burnt down in a devastating fire and the invaluable collection of scrolls was reduced to ashes. In an attempt to win Cleopatra’s heart, Mark Antony bought the entire Library of Pergamos, which had been second only to the Great Library of Alexandria, and presented the 200,000 volumes to Cleopatra. By having done so, Alexandria continued to boast the greatest library in the world until its eradication by Islamic troops entering the city in the 7th century under the leadership of Caliph Omar. The Caliph ordered the destruction of all books on account of either opposing the Koran, which was considered to be heresy or for being superfluous. For years, the water of the public baths of Alexandria was heated using those books to feed the fire. Mathematicians learned their lesson: in order not to disappear, the Brotherhood expanded to everywhere without having a center but its members continued keeping in close contact with each other. This marked the beginning of the mathematical tradition of sharing every doubt, every finding and every gossip with the near and distant colleagues (feedback). Should we call it the first and oldest peer-to-peer network?

Soon two separate branches of mathematics i.e., applied mathematics and number theory began to evolve dividing the mathematician community. Those more intrigued by the practical application formed working groups whereas those merely interested in numbers preferred to work alone. Newton accused the representatives of the latter branch of being vulgar ego jugglers, who had been wasting their time teasing each other with riddles which lacked concrete utility. An ardent supporter of that school was a judge in Toulouse called Pierre de Fermat, whose greatest pleasure in life was to play with numbers and then to send little notes about his findings to Descartes and Pascal. When it came to mental calculations such were his skills that Fermat had no inclination to spend time putting them in writing claiming that it would halt his reasoning and hated being asked about the intermediate steps. He was solution -rather than demonstration- oriented and found it appealing to be compared to big names.

One day he took the famous theorem of Pythagoras (the square of the hypotenuse equals the sum of the squares of the legs) and concluded that the equation has no integer solutions for powers greater than two.

“It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.” he scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus thereby creating one of the most notable puzzles in mathematics driving many mathematicians to insanity until proven three hundred and fifty years later.

The Prize

In the first two hundred and fifty years it was a puzzle merely for young mathematicians who immediately abandoned it as soon as they had arrived to the conclusion, previously drawn by Gauss, that solving this puzzle would not contribute to the progress of mathematics. It was toward the end of the nineteenth century that a German amateur mathematician Paul Wolfskehl struck by love sickness was on the verge of committing suicide.

While waiting for the stroke of midnight to shoot himself in the head he passed the time reading about the latest progress in solving Fermat’s Last Theorem. Touched by a flash of inspiration he began working on the solution and so lost he became in it that morning found him in his library. Even though his approach failed at last the intricate beauty of number theory once again awakened his desire for life.

The young Wolfskehl turned out to have more talent for business than for pure numbers and over the years he had become a millionaire, however, he never forgot that night. In order to repay the debt he rewrote his will bequeathing 100,000 Marks to the first person who could solve the puzzle. Following his death in 1906 the Wolfskehl Prize was announced fueling public desire to prove Fermat’s Last Theorem, which greatly contributed to the progress of number theory.

The Solution

A mountain of failed attempts to win the prize had been accumulating in the basement of the University of Göttingen, 621 proofs, all flawed had been sent just within the first year of Wolfskehl’s death. By 1993 no serious mathematician would engage in trying to solve the longstanding riddle of Fermat’s Last Theorem, only fans, mostly inmates and lunatics, persisted.

Then in 1993 it was at the Isaac Newton Institute in Cambridge, the heart of the world of mathematics (a building created with the intention of bringing together the greatest mathematicians for a week each year, a building with not a single private corner, with offices having no doors and with blackboards even in bathrooms and in the elevator) that a freckled Englishman announced to his illustrious peers that working fully alone for ten years without the aid of a computer he had solved Fermat’s Theorem.

Returning from his classes at Princeton University he would sat alone at the table, sometimes even twelve hours, a piece of paper lying in front of him every so often scribbling a formula like the old Fermat did. However, unlike Fermat he diligently noted down not only each and every formula but their endless developments also.

Mathematicians say their specialty is an archipelago of small certainties scattered in a sea of ignorance. The real advances in mathematics are made when the bridges connecting these islands are built. In his dissertation Andrew Wiles established so many bridges that the entire history of mathematics could be reconstructed based on that, and that is exactly what the Hindu Simon Singh did by writing his beautiful book entitled Fermat’s Last Theorem.

All representatives of number theory are enumerated in that book, however, the most remarkable of all is an unnamed colleague, who when facing Wiles as he was triumphantly leaving the stage addressed him with undisguised obfuscation: “And now that you have taken away the problem, what are you going to give us in return?”.

The New Problem

The secret of Fermat’s Last Theorem had been deciphered at last, however, even as of today the world is abundant in puzzles, which despite the tremendous advances in technology, are still waiting to be solved. All my adult life I seem to have been attracted to problems the complexity of which would deter even the most enthusiastic experts of the field. Inspired by the difficulty of the task I have always managed to rise to the challenge and sparing no time and energy succeeded in unraveling the mystery surrounding it.

After having designed mission critical systems for banks for more than a decade, I found myself in the United Arab Emirates establishing a techno-artistic startup called Nomad Inception boasting a technology which besides having enabled the automated creation of non repetitive Islamic geometric patterns on a scale never seen before also paved the way for new original non repetitive patterns for the first time in five hundred years.

Being Argentinean I have absolutely no relation to the Arabic culture, however, my fascination with the amazing intricacy and sophistication of Islamic geometric patterns helped me grasp the secrets of a traditional art form mastered by a handful of artisans only. Solving a thousand-year-old historical puzzle and thereby potentially contributing to the preservation of our cultural heritage is a vastly rewarding experience worth all the effort invested in it.

It was in Dubai, where I was setting up my company called “Nomad Inception”. There I first learned about bitcoin and having spotted the excellent opportunity to use this innovation for good my new mission was conceived:

Enabling a world where people can freely interact electronically without unnecessary third party interferences. Both for social and commercial interactions. No spying, no censorship, no taking a cut on private transactions between individuals, no mining of private information, no unnecessary middlemen. A world where people are more important than entities like companies and states, a world where people have the choices and the means to interact directly between each other.

To get there we need direct device to device communication, with data being stored at end user devices and apps built to interact with each other directly, over the Internet but without going through the web or requiring any service from any company or institution. We need Person-to-Person Apps that can run independent of any entity. Creation of such a technology is the new problem to solve. The problem is as complex as it can be and technologies like bitcoin where showing the way and later proved to be crucial to find the solution.

That solution involved finding the islands not yet discovered and building bridges to connect them both with each other and with the entire current archipelago.

A Divine Sign

I delved into studying the problem with newfound enthusiasm, however, my commitments at Nomad Inception prevented me from devoting all my attention to finding the solution until a business meeting with a Dubai sheik changed my life forever.

I am anything but superstitious, however, it goes without saying that the question that sheik asked me upon receiving my business card, was that the word “inception” in Arabic appeared as the company logo, could only be interpreted as a divine sign compelling me to leave Nomad Inception without further delay and turn all my attention to find the solution for this modern day Fermat’s problem. You may wonder what the life-changing question was…

“Have you been aware that the word ‘inception’ in Arabic used as your company logo would read ‘bitcoin’ just by adding one dot to it?”

If you are interested in learning about the technology that was created so far after the previous story ended, this list might help you:

1. 
   

   “Fermat, the Internet of People and the Person to Person Economy.”The Internet of People architecture dissected.

2. 
   

   “Introducing the Graphchain.”The cryptographically secured data structure we use to store profiles and their relationships.

3. 
   

   “Introducing Redtooth”Like Bluetooth with global range.

4. 
   

   “The Profile Server.”The cornerstone software of the Internet of people.

5. 
   

   “The Location Based Network.”The geo-located network that help other services to be geo-localized.

The Internet of People is being built by the Fermat Project. If you like what you are reading, consider visiting us at our on-line community on [Slack](mailto:[email protected]?subject=Please invite me to the Fermat Slack Team&body=Hi! I have visited Medium.com and I am interested to join your slack community. Cheers!).

A bit about me: I am a system architect that started his career designing and building banking systems. Later I turned into an entrepreneur. Three years ago I learned about bitcoin and decided I would use that technology to fix the biggest problem we have as humans: “unlimited concentration of power”.