The Prime Number Theorem Trishla Shah Prime Numbers Prime numbers have always been of great fascination to people, whether they were mathematics or not. Nowadays, a prime number is defined as any natural number that can be divided only by 1 and itself and have no remainders. The first few prime numbers are 2, 3, 5, 7, 11. While it is a common misconception that 1 is also a natural number, it is mathematically impossible to have that. 1 is only divisible by 1, but in order to fulfill the requirements, it has to be both divisible by 1 and itself. Itself, being 1, cannot account for both numbers in this case. The largest prime known today has over 17000000 digits. However, we have not always known all this about primes. It seems like such a simple concept to us today, but it has been studied extensively throughout human history in various parts of the world and throughout all time periods. One question that was bound to come up was the idea of continuity. As numbers grow bigger the factors are increasing so the frequency of primes must go down. From 1 to 100 there are 25 primes. However from 101 to 200 there are 21 primes. As numbers get larger, the distance between primes gets larger as well. Do primes go on forever? The answer seems to be yes, but proving that is something entirely different. This is the basis of the Prime Number Theorem. While the first proof of it came out in 1949, this question has intrigued and captivated mathematicians since the beginning of time. The definition of a prime number was defined in the Fundamental Theorem of Arithmetic. While the Theorem comes closely from Euclid's Elements book series. Books VII to IX are about basic results in the theory of numbers. The Fundamental Theorem of Arithmetic is not fully written in the books, however there are two very important propositions that have a very close connection to it. They are known as VII.30 and VII.31 in Book 7. Then IX.14 is a uniqueness theorem in Book 9. The Fundamental Theorem of Arithmetic follows directly from these. VII.30. If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers [i.e., if a prime number c measures ab, then c will measure a or b, where "measure can be translated as "divide, although repeated subtraction would be nearer to the spirit of the Greek word]. VII.31. Any composite number is measured by some prime number. From these two, the Fundamental Theorem of Arithmetic follows. The theorem states that a natural number p > 1 is called a prime number and its only divisors are 1 and itself. It is easily found that 2 is the only even prime in the world, as any other even number would be divisible by 2 as well. It also goes on to say that any natural number that is not prime, such as 4, 6 and 8 is called a composite number. Number are called coprime is the only number that divides the two numbers is 1. Numbers are coprime if the gcf (a:b)=1. One of the most interesting things to come from this, aside from the Prime Number Theorem, is Goldbach's conjecture. Christian Goldbach was a prominent mathematician in the 1700's. He wrote in a letter to Leonhard Euler that ""at least it seems that every number that is greater than 2 is the sum of three primes. Euler went on to rephrase this asserting that " all positive even integers greater than 4 can be expressed as the sum of two primes. This bold claim was left unproven until 1973 by a Chinese mathematician, Chen Jingrun. He even went on to add that any large even number is also at most the sum of a prime and a product of at most two prime numbers. Euclid was also able to prove that if every number in the form of 2n − 1 is a prime number then every number in the form of 2n−1(2n − 1) is a perfect number. With all these advancements and interest in primes, it is clear that they have been studied for a long time. Ever since their first discovery with the ancient Greeks. They are recorded as the first ones because their finding were written down and managed to survive. It is very much possible that there was someone before them, who is unfortunately robbed of all credit. There is some current debate, a dissenting opinion in fact, saying that the ancient Egyptians also discovered and worked with prime numbers. Nonetheless, in Greece there were schools dedicated to the advancement of mathematical knowledge. One of these was where Pythagoras researched. He is well known for the pythagorean theorem, which states that the square of two sides of a right triangle are equal to the square of the third, and longest side. This is a crucial part of mathematics today. However, the greeks themselves were more interested in both perfect numbers and amicable numbers. Perfect number are those that are equal to the sum of all of their factors. The first perfect number is 6, as 1+2+3=6. The next perfect number is 28, and the gap between perfect numbers increases as they grow. Amicable numbers are pairs in which the the sum of the divisors of one equal the second number. The smallest pair is 220 and 284, in which the factors of 220 are 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 and 284, where 1 + 2 + 4 +71 + 142 = 220. These discoveries started to pave the way for the eventual discovery of prime numbers themselves. The next step in this was created by Eratosthenes. It was called the Sieve of Eratosthenes, and it is a way to find every prime in a set of numbers. It is very simple to do, however when the range gets rather large, it becomes a very painstaking task. First, write out every number in the range starting with 2 as the smallest number. For this example we will do from 2 to 50. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Then start with 2, and cross out every second number from then on. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Find the next non-crossed out number, 3 in this case, and cross out every third number from there on. If the number is already crossed out, skip it and go to the next one. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Continue this until there are no numbers left that have multiples left, and all that remains are the list of all the prime numbers in that range. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 While this method works entirely, it is not plausible to keep doing this as the numbers get larger and larger. In fact, as the numbers get larger and larger, testing them to see if they are actually primes becomes more and more time consuming. Pierre de Fermat recognized this and came up with he Little Theorem. It states that p−1 if if there are two numbers a and p, and p is a prime number, then ax p will always have the remainder of 1. This became a crucial tool for future mathematicians to figure out the next so called prime number is 2 actually a prime number. Through all this, there were any efforts to try and find a formula for prime numbers. One of them is as Fermat's primes. He put forth the equation 22n + 1 However, Euler later found that certain numbers in that sequence are actually composite, such as 223 −1. This was a solid blow against the idea that all the numbers in 22n −1 were prime. Next came French mathematician Marin Mersenne. There was another equation during this time period to solve for all primes, but he showed otherwise. He was able to prove that the equation 2n − 1 did not account for every prime number, and in fact was only truly valid between the numbers n=2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. Any n larger than 257 would be composite, not prime. Mersenne primes were names after him where such equation followed 2n − 1 as long as n is a prime number. Next in line was Friedrich Bernhard Riemann in the 19th century. He put forth a hypothesis which was later known as the Riemann hypothesis.