Modern Insights into Ancient Mathematics Debora Mroczek September 2019 When we discuss ancient mathematics, the revolutionary ideas of Thales, Euclid, and Pythagoras, it all seems set in stone, both figuratively and literally. Many of these ideas are so old, that they were actually recorded in stone tablets. From the perspective of modern mathematics, one might wonder, with so many great minds that have devoted entire lifetimes to thinking about the fundamental rules of mathematics, what is even left to think about? What contributions could we possibly make after thousands of years? One could also guess that current research in mathematics happens at much higher level than it did in Ancient Greece, so there would be no point in looking back to Ancient mathematics, except for historical purposes. That is true only to a certain extent. While the tools mathematicians have access to nowadays, such as modern number theory and supercomputers, certainly give them an advantage, they are also in part still trying to understand the same aspects of mathematics as the famous Ancient thinkers. Perhaps the most representative example of this idea is the Pythagorean Theorem. For a right angled triangle, a2 + b2 = c2 | a result commonly used by ninth graders and expert mathematicians alike. The oldest mention to the Pythagorean Theorem dates back to Babylonian tablets from 1600 B.C. In the tablets, there were pictures and examples Figure 1: Babylonian tablet that served effectively as proof of special cases of the theo- from around 1600 B.C. with illustration of special cases of rem [1]. For instance, one of the illustrations proved that the the Pythagorean Theorem. 1 p p p diagonal of a square with sides of length one is 2, which effectively says 12 + 12 = 2. It cannot be determined with much credibility if Pythagoras (570 B.C. - 495 B.C.) or a member of his school was the first person to prove the theorem for the general case. Later on, Euclid (365 B.C. - c.300 B.C), in Elements, provided a proof for the theorem in Proposition I.47. Also, in Proposition I.48., Euclid proved that the theorem is reversible, i.e. the converse is also true. A triangle whose sides satisfy a2 + b2 = c2 is necessarily right angled [2]. Although Euclid was the first to provide a robust approach to both the Pythagorean Theorem and its converse, in the two thousand years that followed, Pythagoras's famous theorem collected over 300 proofs, not only for the theorem itself but also for analogues and generalizations. More precisely, in 1968, there were 367 dif- ferent known proofs for the Pythagorean Theorem and its variations [3]. The fact that hundreds of proofs for the same theorem already exist does not mean that people have stopped trying. In 2015, yet another proof of the Pythagorean Theorem was published [4]. Naturally, one might wonder, what is the point of prov- ing the same thing over and over again? It certainly does not make the results more true. So why are we still trying to shed light on something that was proven to be true two thousand years ago? At this point, even a former U.S. pres- Figure 2: The cracked domino, a rather sim- ident, namely James Garfield, has managed to prove that ple visual proof of the 2 2 2 Pythagorean Theorem by for a right angled triangle, a + b = c . The question of Pakoslaw Gwizdalski. intellectual resources might also come into play | why are people who could be otherwise making new discoveries of possibly much greater impact wasting time on something that has been proven true hundreds of times? Regardless of whether these new proofs are driven by a layman's curiosity or by serious and committed mathematical research, there is value in finding new ways to prove old results. Sometimes the value is purely pedagogical. In the case of James Garfield and Kaushik Basu, author of the 2015 proof, both proofs use different methods and concepts 2 to arrive at the final result. Hence, we learned something new about how these particular methods and concepts can be applied. Perhaps that will inspire another reader to use a similar approach to a different problem. Furthermore, by looking at a problem from many different perspectives we can sometimes find new results. In the case of the Pythagorean Theorem, many analogies and generalizations were found. For instance, the law of cosines and the Euclidean distance in R3, both of which have vast applications across mathematics and physics, were derived directly form the Pythagorean Theorem as generalizations of the original result. No matter how many times something has been proved, we always seem to gain from learning new ways to prove it. Although an interesting case, the Pythagorean Theorem is mostly an outlier. It is not the norm for ancient mathematical ideas to get proved over and over again, hundreds of times. Nor is it the case that a relationship discovered thousands of years ago continues to have an impact in modern mathematics. There is, however, serious research taking place all across the world that aims to refine and deepen our understanding of concepts that have been around since ancient times. Let's take number theory, for instance. The study of integers and their properties. That is perhaps the most intuitive field of mathematics, because it concerns our day-to-day reality. After all, the most basic thing one can do with numbers is to count them. Since ancient times people have separated integers, in particular natural numbers, into categories and tried to establish relationships between groups and categories. Yet, number theory remains an active field of mathematics in the academic environment, with many brilliant mathematicians dedicating their lives to the study of the relationships within and between different types of numbers (odd, even, prime, square, Fibonacci, etc). A fascinating example of how modern number theory continued to develop an ancient concept is the case of perfect numbers. Around 100 A.C., Nicomachus of Gerasa (c.60 - c.120 A.C.), published his first book \Arithmetike eisagoge" (Introduction to Arithmetic), in which he makes the distinction between even and odd numbers, then proceeds to refine this categorization and develops the concept of prime and composite numbers. 3 Nicomachus also introduces the concept relatively prime pairs (numbers that only have 1 as a common divider) [5]. Clearly, Nicomachus's work was a game changer for arithmetic (number theory being its modern equivalent), and we might never understand just how much of an impact his work had at the time. One of the concepts Nicomachus discusses in his book is the idea of perfect numbers. Perfect numbers are numbers which parts add up to the number itself. Nicomachus thought of a number's divisors as its parts. For instance, 6 is a perfect number because 6 is divisible by 1, 2, and 3, and 6 = 1 + 2 + 3. Notice that in Nicomachus's definition, 6 itself is left out, since 6 is the number itself and not one of its parts. Ancient Greeks were able to find four of such numbers: 6, 28 496, and 8128. At the time this seemed to hint at a conjecture. We have one perfect number in the units (6), one in the tens (28), one in the hundreds (496), and one in the thousands (8128). One could find it be reasonable to expect a single perfect number Nk for each group of k digits. In fact, in the 16th century, Girolamo Cardano, one of the greatest mathematicians of that time, carelessly made that conjecture, which now we know is false. In fact, the number of digits increases pretty quickly as we move along the list of perfect numbers. The fifth known perfect number, 33; 550; 336, has 8 digits (a four digit jump), and the ninth on the list accumulates an astounding 37 digits. In his book, Nicomachus provides a procedure for find- ing perfect numbers, which is reproduced in Euclid's Ele- ments as Preposition IX.36, \If the sum of the numbers 1; 2; 4; :::; 2(n−1) is prime, then this sum multiplied by the last term will be perfect." This recipe is not necessarily the only way to obtain perfect numbers, but it is the only known pro- cedure. It might seem like having an expression would have made it easy for mathematicians to produce an extensive list of perfect numbers back then, but as we have seen the num- Figure 3: Nicomachus of Gerasa (c. 100 A.C) [6]. ber of digits starts to increase rapidly as we go along. Before 4 high-performance computers became the norm, we only knew a dozen of these perfect numbers. In 1951, the biggest known perfect number had 77 digits. In 1952, when the next perfect number was found, it became clear why the help of technology was needed | the 13th perfect number has 314 digits. Between 1952 and 2018, 38 more perfect numbers were discovered, the largest of them accumulating a mind-blowing 49,724,095 digits. Once again, one might wonder, why does it matter? While there is no practical application for perfect numbers, the fact that modern mathematicians kept digging at this ancient concept continued to raise questions which answers define the underlying structure of natural numbers. These big questions have opened the door to scientific collaboration and discovery. For instance, the recipe given by Nicomachus in his book also happens to generate prime numbers of the form 2n−1 (commonly refered to as Mersenne primes). The search for Mersenne primes resulted in one of the largest collaborative efforts known to mathematics research, the Great Internet Mersenne Prime Search, which ranked in the top 500 most powerful virtual computer systems in the world in 2012 [7].