Number Theory Sutton Olesen Table of Contents Introduction…………………………………………………………………...Page 3 Chapter 1: Fermat’s Last Theorem…………………………………………...Page 4 Chapter 2: The Twin Prime Conjecture……………………………………..Page 13 Chapter 3: The Riemann Hypothesis………………………………………..Page 20 Chapter 4: Goldbach’s Conjecture…………………………………………..Page 25 Conclusion…………………………………………………………………..Page 30 Works Cited.………………………………………………………………...Page 31 Image Credits………………………………………………………………..Page 33 2 Introduction Since I was born into the world, I was placed in an environment full of numbers, signs, and strange symbols, just begging me to play with them. The world: Number Theory. So when the opportunity of Expert project arose, the choice was obvious. Number theory has captured my attention since I was young, as it amazed me how such simple mathematical problems required complex formulas and complicated math to obtain the answer. How could numbers and formulas change the world the way that they are? Along the process of learning and discovering mathematics, particularly in the past few years, I became more and more absorbed into the subject. I started to want to know the impossible, and asked many questions to almost every teacher I knew. However, all my research beforehand left me with four main pieces of information: my chapters. The four conjectures and theorems I chose to include captivated my urge to learn more. Although some thoughts of mine were left unanswered at the end of my project, I still managed to find some astonishing answers. I will leave the inquiries and solutions for you to consider and discover throughout my paper. Mathematics is one of the most essential concepts to understand, and at the heart of it all lies number theory. Certain problems in number theory have had a major impact in society, and if some are solved, the world will benefit greatly. One must listen carefully in order to comprehend these problems, and thus pave the way for society to be improved. 3 Chapter 1: Fermat’s Last Theorem Fermat’s Last Theorem is: prove that there are no whole number solutions to an + bn = cn where n is greater than two. The tale of Fermat’s Last ​ ​ ​ ​ ​ Theorem begins a long time ago, in 569 B.C. The equation was based off of Pythagoras’ theorem. His theorem was: for all right angled triangles, ​ a2 + b2 = c2 where a is one of the two smaller sides, b is the other small side, ​ ​ ​ ​ and c is the hypotenuse. Pythagoras of Samos (Greece), was one of the greatest ​ ​ ​ mathematicians of his time. Somewhat like fairy tales, there are no personal accounts of Pythagoras’ well-being. All the curious mind can find are stories that have been passed down from generation to generation, so it is hard to know the truth from the false. No matter the way, he created one of the most famous formulas to date. Insight on Pythagoras’ work and life can be obtained by looking at how he studied and gathered his information. He adopted most of his mathematical strategies from the Egyptians and the Babylonians. Both cultures had been exceeding expectations, using complicated equations to make modern accounting systems and breathtaking architecture. It is easy to see why Pythagoras studied these civilizations. Pythagoras noticed that both groups of people only did the formula that always gave the right answer for their problems, not caring about why it worked. Pythagoras wanted to find out why they worked, similar to the way Fermat expanded on Pythagoras’ Theorem much later. Pythagoras continued to travel, gathering students interested in his work along the way. Soon, he settled into a new home, in the town of Croton, Italy, and established the Pythagoras Brotherhood. The group included around 600 followers, 4 who listened and learned from Pythagoras’ lectures, and also created their own conjectures and proofs. Word of the Brotherhood spread, and soon Pythagoras was a mini-celebrity. He became so well-known he even came up with his own word. He created the term “philosopher”. While Pythagoras was watching the Olympics, the Prince of Philus, Leon, questioned Pythagoras. He asked, “What would you call yourself?” Pythagoras replied, “I am a philosopher.” Leon was confused, since he had never heard the word before, and asked what he meant. Pythagoras then proceeded to say: “Life may well be compared with these public games (The Olympics) for in the vast crowd assembled here, some are attracted by the acquisition of gain, others are led on by the hopes and ambitions of fame and glory. But among them, there are a few who have come to observe and to understand all that passes here. It is the same with life. Some are influenced by the love of wealth, while others are blindly led on by the mad fever for power and domination. But the finest type of man gives himself up to discovering the meaning and purpose of life itself. He seeks to uncover the secrets of nature. This is the man I call a philosopher, for although no man is completely wise in all respects, he can love wisdom as the key to nature's secrets.” 5 Pythagoras inhabited much knowledge of everything around him, for he was an observant and wise man. As intelligent as Pythagoras was, he and his Brotherhood had its secrets, as well as some mysterious moments. The Brotherhood was private, and every member swore an oath to always keep their mathematical breakthroughs quiet. If they betrayed this oath, they were subject to death. Furthermore, Pythagoras also died in a very mysterious way. Around 510 B.C., there was a fight in Sybaris, the sister city to Croton (where Pythagoras lived). The fight was between the old government and the new government. The new government won, and the leader of that army, Telys, created an audacious campaign. His goal was to kill all supporters of the old government. Members of the old government and its followers fled to Croton, and Telys ordered all runaways be brought back to Sybaris and be given their punishment. Pythagoras convinced the people to fight back and protect the escapees. This maddened Telys, and he summoned an army of 300,000 men and rushed to Croton, where Pythagoras was waiting with 100,000. Over the span of two and a half months, Pythagoras and his army emerged victorious battle after battle and eventually won this so-called “second” war. Even though the war had ended, Croton remained in trouble. The citizens of Croton worried that Pythagoras would take command of the city. One person in particular, someone by the name of Cylon, rose as the leader of the town. Cylon was evil. One night, Pythagoras’ home and the connected school were circled, the doors and windows were barred, and Cylon set the building on fire. Pythagoras and most of his students were killed. Pythagoras’ story is a strange one, and someone would find it hard to believe that by tweaking one simple number, in a simple equation that Pythagoras created, 6 would make an even more peculiar story. Pierre de Fermat changed Pythagoras’ Theorem without knowing what would happen. Pierre de Fermat was born in Southwest France in the town of Beaumont-de-Lomagne. He was born on August 20th, 1601. His father was a leather merchant by the name of Dominique. This job made a small fortune, and was enough to afford for Pierre to go to extravagant schools. First, he joined the Fransciscan Monastery of Grandselve, and then went to the University of Toulouse. When Pierre was 30, he was made a councillor at the Chamber of Petitions. Councillor’s are the glue connecting the town to Paris, meaning that if a serious case arose, and the central government of France (in Paris) needed to know about it, it was the councillor’s responsibility. He was talked into the political life by his family. Part of his job was to be a judge. Fermat was high in the ranks, and he usually handled cases that were of the more complicated type. Two of his clients, a pair of men named Wallis and Digby, often wrote letters to Fermat. Although Fermat was not necessarily friends with them, these letters show sneak peeks of Fermat's work and life. As time went on, Fermat climbed the ladder of politics and became one of the most powerful men in France. However, he did not move up because he was a good politician, but rather as a result of the plague. Many politicians, including Fermat, stepped up a notch to fill the places of those who died. Fermat was almost replaced himself, as he also caught the virus, and became sick enough that he was pronounced dead for a while. Eventually, the truth came out that he was okay, and everyone let out the breath that they had been holding. After this experience, Fermat was not that interested in politics, and put all of his spare energy in to 7 mathematics. However, math was not a popular archetype at the time, and not many academies offered a course of mathematics. In fact, the only school in Europe that had a class on mathematics was Oxford University, in England. Fermat was a great student, as he was easily teachable. However, he had no one to teach him in his early years. He began to look for a professor, and found a teacher in Father Marin Mersenne, an established mathematician. Mersenne had been traveling through France, spreading news of new findings, when he ran into Fermat. Fermat and Mersenne became good friends, and exchanged letters frequently after Mersenne left for his next obligation. Fermat found many wonders of mathematics under the wing of Mersenne, and Mersenne tried to make Fermat share his magnificent breakthroughs with the community.