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.”

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. Fermat refused, similar to Pythagoras and his Brotherhood. Fermat continued to find many theorems and proofs, and rumors began to spread that he was an accomplished mathematician. People began to want to see his work. Again, Fermat was secretive, and he tricked them. He did this because most mathematicians at the time were English, and Fermat was French. During Fermat’s lifespan, France did not particularly like England. When he was asked to share his work, Fermat sent letters to mathematicians with a theorem that he proved, but he did not include the proof. Fermat was also a founding father of mathematics, doing so by helping make two areas in math, calculus and probability theory. By most modern day mathematicians, probability theory is defined as, “a branch of mathematics that deals with quantities that are random,” and calculus is defined as “the ability to calculate the rate of change of one

8 quantity to another.” At first, Isaac Newton took all the credit for discovering calculus. That all changed when Louis Trenhard Moore found a piece of Newton’s work in 1934. The paper read that Newton’s field of calculus was based on Fermat’s way of drawing tangents. Fermat found many mathematical wonders in his lifetime, but perhaps the most famous one was his last. Fermat created the problem: Prove that there are no whole number solutions to an + bn = cn where n is greater than two. He then spent years trying to find if it was true or not, and in 1637, claimed that he had proved it, saying, “I have discovered a truly remarkable proof of this theorem, of which this margin is too small to contain.” However, before he could show his work, he died on January 12th, 1665, in Castres, France. In his time on Earth, Fermat produced and proved many theorems. Fermat’s Last Theorem is one of the most famous problems ever to exist, and the mathematics behind it makes it even more fascinating. Fermat’s Last Theorem is so remarkably hard, it extends to other problems in mathematics. There is another problem related to Fermat’s Last Theorem. It is called the Taniyama-Shimura Conjecture. It states: Elliptic curves over the field of rational numbers are related to modular forms. It is not needed to understand this problem. Elliptic curves are from the field of geometry, and modular forms are from a field of mathematics called analytic number theory. In mathematics, when two areas of mathematics are joined in a problem, it is monumental. The Taniyama-Shimura Conjecture was also crucial to mathematics because it is related to Fermat’s Last Theorem, because if the Taniyama-Shimura Conjecture is proven true, Fermat’s Last Theorem would in result also be true. Richard Frey proved the relationship between these two problems. Even though the Taniyama-Shimura Conjecture is important in mathematics, the legacy of Fermat’s Last Theorem is

9 much more important to mathematicians, and many mathematicians have attempted to solve it. However, only one person could be the first to solve it. Fast forward 300 years to the mid 1900s, where Andrew Wiles came into the world. Wiles discovered Fermat’s Last Theorem at the mere age of ten. Wiles was at a small library on Milton Road, near where he grew up. He was searching the shelf for a good book, when he came across The Last Problem, by Eric Temple Bell. He was young, but he was immediately hooked. Wiles grew up to be one of the greatest mathematicians of our time. He is an immigrant from England who found a job at Princeton University. At the time, no one had been able to prove Fermat’s Last Theorem, but if anyone could prove it, it was Andrew Wiles. Wiles’ life goal became to solve Fermat’s Last Theorem, just like many other mathematicians. However, Wiles was different than those other mathematicians, and he defied the beliefs of mathematicians while attempting to solve Fermat’s Last Theorem.“Young men should prove theorems, old men should write books,” a mathematician named G.H. Hardy said in his book A Mathematician’s Apology. Also, in mathematics, groups of people normally work together on problems, and therefore share the credit if they solve it. The universe of math is similar to a giant group chat, where everyone can share their discoveries. Despite proving theorems usually being a younger person’s job, Wiles, from ages 33-40, worked on Fermat’s Last Theorem in total isolation. When people thought he had stopped, he was creating unheard of ways to tackle the problem. Wiles spent seven years trying to solve Fermat’s Last Theorem, and after long, hard, and tedious work, he succeeded.

10 Normally, if someone accomplishes an enormous feat, like solving Fermat’s Last Theorem, they become famous, assuming it is correct. As soon as Wiles announced his completion, the world went crazy. He quickly became famous. So rapidly, in fact, he was even named one of the most influential people in the world by People Magazine in 1993, when he originally solved it. However, the proof still had to be checked to be sure it was accurate. “Referees” as they call them, are the mathematicians who check proofs. The proof of Fermat’s Last Theorem was 200 pages long and divided into six chapters. Per usual, there are only three referees assigned to a proof, but obviously Fermat’s Last Theorem is anything but normal. There were six referees assigned, one for each chapter. One of the referees who checked Wiles’ proof was named Nick Katz. In chapter three, he found a very serious flaw. Katz could not tell anyone but Wiles about the mistake, as referees are sworn to secrecy. While Katz alerted Wiles immediately, the media wanted to see the proof. Obviously, Wiles did not release the proof, since he was now aware of the mistake. He worked desperately to fix it for a year, but to no avail. The media started to spread rumors. Wiles had revolutionized mathematics by “solving” Fermat’s Last Theorem, only to have it be false. Wiles was devastated. After all this commotion, the media was anxious for Wiles’ proof, and they started to give up on Wiles. After some time, Wiles told the world about the flaw. He was embarrassed, but still would not show the world his proof. Wiles biggest fear was that someone would fix the error and take the credit from him, as there was a million dollar prize. Andrew wanted all the credit. Eventually, Wiles sought some help in the form of one of his closest friends, Richard Taylor. By this time, the world was anxious enough that someone from the mathematic community faked a solution to an + bn = cn where n is greater than

11 two. Eventually, Wiles figured out the solution was false, and resumed his work on Fermat’s Last Theorem. After this drama, it was now 1996, and Wiles had the spotlight back on him. With all the attention, he gave himself a one month deadline to solve the problem. Almost at the end of his deadline, Wiles was able to overcome the challenge, and solved the Taniyama-Shimura Conjecture, thus solving Fermat’s Last Theorem. However, not everyone was so sure. After all, Wiles had said the same thing before, only to be false. But, Wiles himself was confident, and he took the opportunity to lecture his proof at Cambridge University without even checking his work. Wiles decided to present his proof over three lectures. After the first, many were confident that the proof was correct, some even betting money that by the end of the week, Fermat’s Last Theorem would be proven true. By the time the third and final lecture had ended, people knew they had just been in the presence of a historical moment. This time, the referees found no mistake, and Fermat’s Last Theorem was solved. “I think I’ll stop here.” - Andrew Wiles, 1996.

12 Chapter 2: The Twin Prime Conjecture

Although the Twin Prime Conjecture may not be as well known as Fermat’s Last Theorem, it remains one of the greatest mathematical conjectures. Before diving further into this topic, it is important to understand some basic knowledge of the conjecture. The Twin Prime Conjecture is: There are infinite pairs of twin primes. A twin prime is two primes separated by only two numbers. They cannot be separated by only one, since that would leave an even number and an odd number, and even numbers cannot be prime. For example, a twin prime is 11 and 13, because they are both prime numbers that are separated by two. Another pair is 101 and 103. The gap between two twin primes seems to lengthen as the numbers become larger, but does it get to a point where there can be no more twin primes? Or are there an infinite amount of them? That is the Twin Prime Conjecture. It is fair to say that if there is not an infinite amount of primes, there cannot be an infinite amount of twin primes. Luckily, Euclid of Alexandria, in 300 B.C., proved that there are an infinite amount of primes. He proved this in his series of 13 books, The Elements. Many mathematicians say that this particular proof is “beautiful.” Euclid's proof is by contradiction. That means that he made an assumption that is opposite of what he wanted to prove, and then found that that assumption led to an absurd statement. Therefore, the assumption was false. The opposite of what he wanted to prove was false, so what he wanted to prove is true. For example, someone is trying to prove that there are an infinite amount of numbers. They would state that there is a number (n) that is the biggest number, and that means that n+1 is less than n. This obviously is untrue, which proves there are an infinite amount of numbers. In Euclid’s case, he was trying to show that

13 there were an infinite amount of primes. Before anything, Euclid found a formula that let him find a bigger prime number if he had a prime number. To begin the proof by contradiction, Euclid assumed there was a prime number that was the biggest prime number (p). However, that means that if he did his formula on p, he now had a bigger prime number than the biggest prime number. This is absurd, and therefore there are an infinite amount of primes. The basic mathematical concept of the Twin Prime Conjecture is within reason to grasp, and looking at the historical side increases knowledge on the Twin Prime Conjecture. The history of the Twin Prime Conjecture is controversial. No one knows much about it for certain. Some mathematicians give the credit of the creation of the Twin Prime Conjecture to Alphonse de Polignac. Born in France on April 27th, 1826, Polignac went to a school called Ecole Polytechnique, and was a second lieutenant in the French Artillery. On October 15th, 1849, Polignac published a paper titled New Research on Prime Numbers. He published it in the Comptes Rendus, a book made by the French Academy of Sciences. Inside of the paper, Polignac wrote many conjectures on the topic of prime numbers. One of them was: Every even number is equal to the difference of two prime numbers in an infinitude of ways. Polignac was simply saying that there are an infinite amount of primes that are separated by an even number (e). If e is a large number, it is easier to prove than if it was smaller. Further into the paper the reader will learn more about this. By an infinitude of ways means that the Twin Prime Conjecture is true

14 for all even numbers, not just two. It works for an infinite amount of even numbers. Polignac’s statement is similar to the Twin Prime Conjecture. If e is 90, then it is easy to find an example, such as 11, 101. Are there an infinite amount of primes that differ by 90? Thirty? Ten? Two? Polignac is saying the Twin Prime Conjecture works for an infinite amount of even numbers. However, the Twin Prime Conjecture only wants to prove e for two. Polignac is not the most trustworthy person, and this is shown by another conjecture in the same book, Comptes Rendus. In this second conjecture, Polignac states: Every odd number is equal to a power of two plus a prime number. (25 is 23 + 17, 27 is 24 + 11.) He claimed it was true for all numbers up to three million. But when Leonhard Euler stumbled across this, he saw that Polignac was incorrect. He found at least two mistakes, 127 and 959. Polignac still tried to come up with an excuse for the falsity, saying other obligations left him with little time to check his work. He stated that he did not do all the math himself, and that the people that had helped him must have messed up his work. It might be true that Polignac jumps to conclusions sometimes, and it is important to understand if this is the case for his first conjecture. So far, no one has proved him wrong nor correct. Even though Polignac wrote something almost identical to the Twin Prime Conjecture, some mathematicians believe another man came up with it. That is where James Whitbread Lee Glaisher comes in. He was born on November 5th, 1848, in Lewisham, London, and died on December 7th, 1928. He went to Trinity College in Cambridge and later joined the Royal Society, a group of intelligent men. He produced many papers on mathematics in his time on Earth. Glaisher also edited the Messenger of Mathematics, an annual collection of math, for 56 years. One year, in 1879, Glaisher added his own paper, An Enumeration of Prime Pairs,

15 into the collection. In the paper, Glaisher defined a “prime pair” as two primes differed by only one number. Twin primes. He then proceeded to list the “prime pairs” up to two million. Finally, he wrote, “There can be little or no doubt that the number of ‘prime pairs’ is unlimited; but it would be interesting, though probably not easy, to prove this.” This is much closer than Polignac, so maybe Glaisher deserves the credit for creating the Twin Prime Conjecture. The history of the Twin Prime Conjecture is debatable, but what about the complicated mathematics? Many interesting revelations about twin primes have been found, each one crazier than the last. One of them involves “finite” numbers and “infinity”. Finite numbers are the numbers with which humans are accustomed. Five, 100, 2,446, 10,476,723, etc, are all finite numbers. All real numbers are finite. “Infinity” is harder to explain, as there is no number than infinity stands for. Generally speaking, infinity is thought to be the biggest number. This cannot be possible, as shown earlier. This is only part of the twisted logic of infinity. A kooky phenomenon is that when you add an infinite amount of numbers, the answer is sometimes infinity, meaning it keeps growing forever, and sometimes the answer is finite. For instance, 1+2+3+4+5+6+ . . . = ∞ (∞ is the sign for infinity) but 1+1/2+1/4+1/8+1/16+ 1/2n ≈ 2 ( ≈ 2 means it gets closer and closer to two but never quite reaches it.) To further understand prime numbers, mathematicians attempted the problem 1+1/3 +1/5+1/7+1/11+ . . . and found the answer to be ∞. Furthermore, mathematicians tried to do this with twin primes: 1+1/3+1/5+1/5+1/7+1/11+1/13+ . . . If this series of numbers equals infinity, The Twin Prime Conjecture would be solved. In reality, however, the answer is finite. The exact answer is unknown, but most mathematicians believe it falls around 1.9. This number is called Brun’s Constant. Though Brun’s Constant is not ∞, it does

16 not necessarily mean the Twin Prime Conjecture is false. Brun’s Constant only adds to the mystery that is the Twin Prime Conjecture. Since Brun’s Constant was found, mathematicians have attempted two different ways to solve the Twin Prime Conjecture. The first of the two comes from Will Sawin and Mark Shusterman. The two of them proved the Twin Prime Conjecture for a related mathematical setting. The related setting involves polynomials. A polynomial is an expression that does not have a negative or fractional exponent, or a variable that is multiplied by less than one (e.g. 0.4x). To illustrate, x + 2 is a polynomial. So is x5 + (4x + 5), and so is 6. Polynomials are categorized by their “order”. Their order is found by the highest exponent. In the polynomial x5 + (4x + 5) , the order is five. In x + 2, the order is one. Polynomials can be factored, just like the way integers can. They can be factored by breaking the polynomial into two polynomials of lesser order. For instance, x2 − 3x + 2 = (x − 1)(x − 2) . If the math is done, it is clear to see that they are the same. The order in the first polynomial was two, and the two split polynomials had an order of one. Similar to integers, when a polynomial cannot be factored, it is considered a prime polynomial. An example of a prime polynomial is x2 + 1 , since it cannot be factored. A more complicated example is 2x2 + 14x + 3 . There are also twin prime polynomials. It requires complex math to see if two prime polynomials are twin prime polynomials, and it is not important to know of this complicated math. It is important, however, to understand that they exist. The two men, Sawin and Shusterman, proved that there is an infinite amount of twin prime polynomials. This result is a huge milestone, and when the Twin Prime Conjecture is solved, if it ever is, twin prime polynomials will probably squeak into the proof.

17 The latter of the two ways to attempt the problem is the more traditional way. It involves e, the variable that appeared in Polignac’s Conjecture. As a reminder, e is an even number separating two primes. Again, e is easier to prove larger than smaller. For the Twin Prime Conjecture to be solved e must be proved to be two. By the year 2013, some had given up on the Twin Prime Conjecture. But not a lecturer at the University of New Hampshire, Yitang Zhang. While Zhang’s proof for e is far from e being two, it is still a monumental breakthrough. He proved e for 70,000,000. Even though 70,000,000 might seem large at first glance, it is microscopic compared to infinity. Right after Zhang released his proof, mathematicians applied the same strategies Zhang used to try to reduce 70,000,000 to a smaller number. They were successful, and the next step down was 60,000,000. Almost instantly after this was proven, the world of mathematics took a giant leap forward, narrowing e down to 5,000. Another addition to the numerous breakthroughs on the problem was contributed by James Maynard. He is a post-doctoral researcher at the University of Montreal. He pushed e all the way down to 600. The most recent addition to the problem was contributed by Terence Tao, a mathematical phenom making huge progress in multiple areas of mathematics. When Tao disappears from this world, he will be as big as Fermat, Riemann, Euler, and the other great mathematicians. Tao proved e for 246. In order for mathematics to be comprehended completely, mathematicians must solve the complicated problem, can e be pushed down to two by the power of modern mathematics, thus solving the Twin Prime Conjecture? Or perhaps a new field of

18 mathematics must be discovered to solve the Twin Prime Conjecture. Is it even mathematically possible for e to be two? That is the Twin Prime Conjecture.

19 Chapter 3: The Riemann Hypothesis

Much like all the other problems, the history of The Riemann Hypothesis amplifies the knowledge of the problem. The Riemann Hypothesis is hard to grasp: All zeroes of the zeta function have real-part one-half, but perhaps the history will help simplify the conjecture. On September 17th, 1826, Georg Friedrich Bernhard Riemann was born. He entered the world in The Kingdom of Hanover, which was part of Old West Germany. There were six children in his family, but only him and a sibling lived to adulthood. Riemann’s father was a veteran of the Napoleon years, and then became a Lutheran minister. Sadly, Riemann’s mother died young. When Riemann was still a baby, his family moved to a quiet village called Quickborn, as his dad had found a new job at the local church. Details of his education are hard to find, and the only resource of information is a memoir, written by Riemann’s friend Richard Dedekind. Before he started official school, Riemann participated in his father’s lessons, similar to a homeschool. Riemann began his formal education at the age of 14. He attended the Gymnasium School, 80 miles away from Quickborn. He went to that particular school because one of his grandmothers lived in Hanover, (where the school was located) so his family did not have to pay boarding fees. Riemann hated the school, and was relieved when his grandmother died in 1742, and he moved to a new school in Lüneberg. However, Riemann did not find success in Lüneberg either, and bounced around to different schools until he landed at the

20 University of Göttingen. His father planned for him to become a minister. At first, Riemann had bad feelings toward Göttingen, but then he learned that Carl Friendrich Gauss lived in Göttingen. Gauss was one of the greatest mathematicians of all time. Although Gauss was 69, and found little fun in teaching, he still lectured sometimes and Riemann attended most of them. It was at Göttingen that Riemann’s passion for mathematics grew. He stopped learning about religion and focused on mathematics instead. Riemann was now engrossed in mathematics, but he was only a beginner. Even with all the success he had had with his education thus far, Riemann still wanted more. He left Göttingen for Berlin University. He spent two years in Berlin, and became the mathematician he is known to be. After this little stunt in Berlin, Riemann went back to Göttingen in 1849, and obtained more knowledge about mathematics. He received his doctorate degree in mathematics in 1851, and in 1854, moved on to become a lecturer. Finally, in 1857, he was made an associate professor at Göttingen. Riemann continued working, and climbed the ranks at Göttingen, becoming a full professor. In 1862, Riemann began to settle into a family, marrying a friend of his older sister, Elise Koch. Riemann died not much later. Only four years after marrying, he died on July 20th, 1866. Riemann died as a wonder of a mathematician, having produced many papers and theorems as well as exploring many fields of mathematics. Although Riemann produced many ideas on mathematics, his one main goal was to better understand the distribution of prime numbers. The Riemann Hypothesis was only an attempt to comprehend the distribution of primes. Contradictory to the first thought, the distribution of primes does not locate where primes are in the sea of numbers that there are, but rather how many primes

21 there exist between two numbers. For instance, the distribution of primes between ten and twenty is four: 11, 13, 17, and 19. The distribution of primes is related to an expression written as π(N). In this case, π does not represent the famous number 3.14159 . . . π essentially means primes. π(N) is calculating the amount of primes up to a given number, (N), similar to the distribution of primes. π(N) grows larger as N becomes bigger. This is trivial. However, the rate of which π(N) increases slows down as the value of N rises. It is simple to discover that π(N) is related to the distribution of primes. If looked at closely, the distribution of primes between two numbers, (x and y), can be found by calculating π(N). By finding π(N) for x, then determining π(N) for y, and then subtracting π(N) of y from the π(N) of x, the answer shows the distribution of primes between x and y. However, identifying π(N) for large numbers can be a tedious task. Many mathematicians have wondered if there was a formula for finding π(N), and worked hard to find it. The first formula discovered for revealing π(N) for large numbers was the Prime Number Theorem. This formula was introduced by Adrien-Marie Legendrien in 1798. The Prime Number Theorem is, π(N) ≈ N . This is saying that π(N) is logN almost equivalent to N . Log is a complicated logN mathematical term that is not necessary to understand in order to grasp the Prime Number Theorem. The Prime Number Theorem provides an approximation of π(N).

22 Again, an approximate. The estimate given for π(N) by the Prime Number Theorem becomes more accurate as N approaches infinity. Knowing this, the guess of π(N) from the Prime Number Theorem for 1000 is more accurate than the estimate for 100. Nearly 100 years later after it was created, in 1896, The Prime Number Theorem was proved by two separate mathematicians, Jacques Hadamard and Charles de la Vallée Poussin. They produced two different proofs. Poussin’s proof received more credit, but Hadamard’s was easier to understand. There was little to no controversy on who deserved the full honor. In both Hadamard’s and Poussin’s proofs, they both involved the Zeta Function.

The Zeta Function is, ζ(s) = ∑ n−s , which is not easy to understand. n However, it is again not required to understand what it means, only to know that it exists. A simpler way to express the Zeta Function is ζ(s) = 1 + 1/2s + 1/3s + 1/4s + 1/5s . . . Where s is a number. For example if s is two, the zeta function is ζ(s) = 1 + 1/22 + 1/32 + 1/42 + 1/52 . . . This is hard to calculate. Riemann wanted to use the zeta function to find a pattern in the distribution of primes, and π(N). In 1857, Riemann produced a paper titled Theories of Abelian Functions. In it, he related the Zeta Function to π(N). Riemann showed that if there is a certain pattern in the zeta function, π(N) can be found. All non-trivial zeroes of the zeta function have real part one-half. This is complicated, and the best way to explain it is to go one step at a time. A zero of a function is a number that is plugged in for a variable in order for the function to equal zero. To illustrate, a zero of the function y = 1/2 x − 2 is four. (Half of four is two, two minus two is zero.) When a number is plugged into the zeta function, sometimes it equals zero, thus being a zero of the zeta function. A non-trivial zero

23 means all zeroes of the zeta function that are not negative even numbers. It has been proved that all negative even numbers are zeroes of the zeta function, and therefore these are trivial zeroes. Furthermore, the non-trivial zeroes of the zeta function are zeroes of the function that are not negative even letters. When Riemann wrote “real part,” he was referring to something known as complex numbers. Complex numbers have two parts. The real part and the imaginary part. The real part can be any number such as one, two, three, -1, -2, -3, and so on. The imaginary part is the letter i multiplied by a number. Every number has an imaginary number and a real part. It is not important to understand how to find them or what they are. Riemann stated that all non-trivial zeroes of the zeta function have a real part of one-half. For example, if four was a zero of the zeta function, even though it is not, Riemann hypothesized that it would possess a real part of one-half. It does not have a real part of one half, but that is beside the point. If it is proven that all zeroes of the zeta function have real part one-half, π(N) would be understood, and the Riemann Hypothesis would be proven true. Since the Riemann Hypothesis is one of the most important problems for mathematicians, many have worked hard on the problem. Some have even claimed to have solved it. The most recent example of this is given by Sir Michael Atiyah, back in 2018. Atiyah was 90 at the time, and was well-known in the mathematics community, as he has solved quite a few theorems. However, his last piece of work was not as memorable as his previous ones, as his claimed proof of the Riemann Hypothesis came nowhere near the answer. In fact, some mathematicians say that Atiyah may have suffered from dementia in his last years.Many others have attempted to solve the Riemann Hypothesis, but none have come close to understanding the holy grail of modern mathematics.

24 Chapter 4: Goldbach’s Conjecture

Goldbach’s Conjecture is: All even integers greater than four can be written as the sum of two primes. Similar to other problems, the historic portion of the story of Goldbach’s Conjecture is vital to understand. Although the mathematics of Goldbach’s Conjecture may be understandable, the people involved in its history enhance the problem. Christian Goldbach was birthed into the world on March 18th, 1690, in Käliningrad, Prussia. His father was a Protestant church minister at a local place of worship. Not much is known about his childhood, but his college and adult years are better documented. Even though he ended up as a phenomenal mathematician, he studied law and medicine in college. After his formal education, he traveled around Europe, and ran into several masters of mathematics along his journeys. Among the mathematicians he met were Gottfried Leibniz, Nicolaus Bernoulli, Nicolaus Bernoulli Ⅱ, Jacob Bernoulli, and Daniel Bernoulli. These mathematicians fascinated Goldbach, and he began studying mathematics. Goldbach went on to create one of the most famous problems to exist. Looking at how he became an accomplished mathematician as well as when and how he introduced his famous conjecture, adds depth to his achievements. In 1720, Goldbach published his first paper related to mathematics, and nearing 1724, he was already well known in the world of mathematics. In 1725, he was announced as a professor of mathematics and a historian at St. Petersburg Academy of

25 Sciences. Goldbach then had the ability to contact other accomplished mathematicians, as he was well-respected. Soon after, in 1728, Russia took over his home-country (Prussia). Goldbach was hired as a tutor to the young Tsar Peter Ⅱ, and in order to carry out his new job, he had to move to Petersburg, Russia. Coincidentally, Leonard Euler, a phenomenal mathematician from Switzerland, also moved to Petersburg in 1727. Goldbach knew this, and contacted Euler. Over time, the two began a long-lasting relationship, and later exchanged letters until death. Goldbach was now in contact with one of the greats of mathematics, Euler, and he wanted to show off his best work to Euler. On June 7th of 1742, Goldbach sent a letter to Euler. In it he posed two conjectures. The latter of the two is known as the Weak Goldbach’s Conjecture. The Weak Goldbach’s Conjecture states: All odd numbers less than or equal to seven can be expressed as the sum of three primes. Just like the Twin Prime Conjecture, breakthroughs on this conjecture come one by one, each closer to the solution than the last. The first breakthrough came from two mathematicians named Hardy and Littlewood. In 1923, the pair showed that every sufficiently large odd number is the sum of three primes, assuming the Riemann Hypothesis is true. Sufficiently large is a bit hard to explain. In mathematics, it means that there is a certain number (n) that beyond n, all numbers are true for the conjecture that is being proven. However, no one knows the value of n. In Hardy and Littlewood’s case, they proved that passed n, all primes are the sum of three primes. In 1937, a Russian mathematician by the name of Vinogradov concluded that all sufficiently large odd numbers are the sum of three primes. In this proof, the Riemann Hypothesis does not have to be true for it to be correct. Vinogradov improved what Hardy and Littlewood conjectured.

26 Next, in 1956, a man that went by the name Borozdkin, proved that

15 Vinogradov’s result was true for numbers bigger that 33 . He gave an exact number for the sufficiently large number, n. Furthermore, Borozdkin knew that n 14 15 could also be smaller, say 33 , but he knew n was at most 33 . It is crucial to understand that if n is proven to be seven, the Weak Goldbach’s Conjecture is proven true. The next addition to the story of the Weak Goldbach Conjecture came in 1995, when a mathematician named Kaniecki demonstrated that every odd number is the sum of at most five primes, again assuming the Riemann Hypothesis is true. He is saying that every odd number can be the sum of one prime, or two primes, three, but at most five. Again, this is only if the Riemann Hypothesis is true. Following Kaniecki, in 1997, a man named Deshouillers and his co-workers displayed that the Weak Goldbach’s Conjecture is true, assuming the Riemann Hypothesis is accurate. Finally, more than 200 years later than when Goldbach first introduced the problem, Harald Helfgott overcame the challenge of the Weak Goldbach’s Conjecture in 2013. Of course, the solving of The Weak Goldbach’s Conjecture did not impact the first problem in Goldbach’s 1742 letter. The problem: Goldbach’s Conjecture. Can every even number greater than two be expressed as the sum of two primes? Much like the Twin Prime Conjecture, almost all mathematicians believe the statement is true, but there is no proof of it. Goldbach’s Conjecture has been upheld for numbers up to 2 · 1017 , thanks to modern-day technology. But even though 2 · 1017 seems like a large number, it is nothing compared to the array of infinite numbers in existence.

27 Once again, the breakthroughs on Goldbach’s Conjecture progress little by little towards the answer. The first occurred nearly 200 years after it was created, in 1939, when Schnirelman discovered that every even number greater than two can be written as the sum of no more than 300,000 primes. Some numbers could be the sum of four primes, others 100,243, some 300,000. The limit that Schnirelman found was 300,000. Following this, in 1975, Montgomery and Vaughn proved that “most” even numbers are the sum of two primes. More precisely, they said that if someone gave them a number (y), they could show that every even number less than y equals the sum of two primes, with a few possible exceptions. In other words, they could say how many “problem children” there are up to y. The problem children could work, just not definitively. However, the two of them could not identify what numbers were the problem children. In 1996, mathematics progressed greatly. Ramaré improved Schnirelman’s threshold from 300,000 to six. It is important to know that, because if this threshold is lowered to two, Goldbach’s Conjecture is true. This is because prime numbers are odd. Furthermore, if the threshold is two, that means that every even number can be written as the sum of either one prime or two. But, it is impossible to express an even number as the sum of one prime, as primes are all odd. Thus, if the limit is taken down to two, every even number greater than two has to be the sum of two primes, since no even number can be odd. Ramaré’s proof is the most recent progress on Goldbach’s Conjecture. Hopefully, many more breakthroughs are coming, but for now, the world of mathematics is not content. Goldbach’s Conjecture is significant enough to the world of mathematics that one of the greatest mathematicians who has ever lived, G.H. Hardy, stated, “Goldbach’s Conjecture is not only the most famous and difficult problem in number theory, but the whole of mathematics.” This may be a

28 little audacious, but nonetheless demonstrates the magnitude of Goldbach's Conjecture. Despite how perfect Goldbach’s Conjecture may seem, every great problem comes with some controversy. When talking about this problem, the argument that arises is whether or not Goldbach’s Conjecture is able to be proven. This does not mean that the conjecture is false, only that it may not be possible to prove it. Prior to 1931, mathematicians thought that mathematics was complete, meaning that every conjecture or theory had either a proof or a disproof. But Kurt Gödel had other ideas. Gödel was a professor of mathematics at the University of Vienna for most of his life, who provided mathematics with a vital bit of information. That important piece appeared in a paper he published in 1931, titled Über formal unentscheidbar e Sätze der Principia Mathematica und verwandter Systeme. This translates to: On Formally Undecidable Propositions of Mathematics and Related Systems. In this paper, Gödel threw off the idea of mathematics being complete, blindsiding almost all mathematicians of his era. To be concise, Gödel proved that every system of mathematics (e.g. Algebra, Calculus, Geometry) contains unprovable truths of mathematics. This means that everywhere in mathematics, there are conjectures that are true, but there is no proof for it. Some mathematicians believe this is the case for Goldbach’s Conjecture. Gödel also showed that it is impossible to know if a problem is unprovable. If Goldbach’s Conjecture is unprovable, no one will ever know. Goldbach’s Conjecture is one of those problems that could be possible, could be not, could be very close to being

29 solved, could be miles away, but nevertheless remains one of the greatest problems in mathematics.

Conclusion

My year can be defined by one word: Expert. I still cannot overcome the fact that I completed the project. Although I found a lot of information on number theory, my research is nothing compared to all the papers out in the world. The amount of published pieces of work on this topic demonstrates the importance of number theory to the world. In fact, one of my most monumental questions was on how number theory relates to the world. This project has impacted me greatly over the past year, in many different ways. The adjustments I had to make changed my life in a positive way, contradictory to what I thought at the beginning of the year. The project is a one in a lifetime adventure, and will live with me forever. In my eyes, Expert was a way to express my interest and show the hard work I accomplished. Even though Expert is over, I wish I could do it again. Sixth grade was one of my favorite years, partly thanks to Expert. Number theory for me is my favorite topic, and I would not change a thing about what I did for my project, except focus on different theories and conjectures. Judging by the amount of fun I had during this span of ten months, I will almost definitely create papers on mathematics in my free time. This project influenced my life in great ways. Go Expert!

31 Works Cited

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Derbyshire, John. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Penguin Group, 2003.

Dunham, William. A Note on the Origin of the Twin Prime Conjecture. https://www.intlpress.com. Accessed 1 Dec. 2019

Friedberg, Richard. An Adventurer’s Guide to Number Theory. McGraw-Hill, 1968.

Hartnett, Kevin. “Big Question About Primes Proved in Small Number Systems.” Quanta Magazine, https://www.quantamagazine.org. Accessed 1 Jan. 2020.

“Mathematician Claims Proof of 159-Year-Old Riemann Hypothesis.” HowStuffWorks, https://science.howstuffworks.com. Accessed 26 Nov. 2019.

Numberphile. YouTube. https://www.youtube.com. Accessed 2 Dec. 2019.

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Numberphile. YouTube. https://www.youtube.com. Accessed 20 Nov. 2019.

Numberphile. YouTube. https://www.youtube.com. Accessed 5 Jan. 2020.

O’Shea, Owen. The Call of the Primes: Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics. Prometheus Books, 2016.

Ross, William. Professor of Mathematics, University of Richmond. 10 Jan. 2020.

Singh, Simon. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. First Anchor books edition, Anchor Books, 1998.

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33 Image Credits

https://www.businessinsider.com https://mathenchant.wordpress.com https://altexploit.wordpress.com https://kids.britannica.com/students/assembly/view/57048 https://www.math.utah.edu/~pa/math/q2.html https://nisciencefestival.com/2018/event.php?e=211 https://en.wikipedia.org/wiki/Bernhard_Riemann https://explainingscience.org/2019/09/01/the-goldbach-conjecture/ https://mathworld.wolfram.com/GoldbachConjecture.html https://www.abebooks.co.uk/first-edition/formal-unentscheidbare