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The Prime Number Theorem Farah Salim The Prime Number Theorem Farah Salim History Throughout history, the concept of prime numbers has fascinated and been the most discussed topic in math- ematics. A prime number is defined as a natural number that can only be divisible by 1 and itself. Some examples include 2, 3, and 5. Many have made the mistake of counting 1 as a prime number, but in reality it cannot be. 1 is only divisible by itself, and as we stated, a prime number is divisible by two natural numbers. This concept may seem simple, but it has caused many mathematicians throughout history to study and understand these numbers. The Prime Number Theorem has to be one of the most fascinating topic within the Mathematical world. Many mathematicians have sought to prove and understand the concept of prime numbers for years. The fact that there are infinitely many prime numbers has not helped the curiosity of others. Today, the largest prime number known has 17,425,170 digits. We obtain the definition of primes from The Fundamental Theorem of Arithmetic. In the theorem it states that a natural number p > 1 is called a prime number and its only divisors are 1 and itself. The most fascinating part of prime numbers are that they are all odd except for 2. All natural numbers greater than 1 that are not prime are called composite. Numbers a; b are called coprime if the gcd(a:b) = 1. Christian Goldbach, a Prussian mathematician in the 18th century, proposed a conjecture that states 'every even number greater than 2 is a sum of two prime numbers'. It was not until 1973 when Chinese mathematician proved Goldbach's conjecture and added that every large even natural number is at most the sum of a prime and a product of at most two prime numbers. The first known civilization to come across prime numbers were the ancient Greeks. This is due to the fact their findings were recorded. However, some historians suggest that the ancient Egyptians had an idea of prime numbers. The ancient Greek mathematicians of Pythagora's school were interested in finding out more about perfect and amicable numbers. A perfect number is one whose divisors are the sum of the number itself and an amicable number are a pair of numbers whose divisors are the sum of the other number. The discoveries made about perfect and amicable numbers paved a way for prime numbers to be established. The first known account of prime numbers is attributed to Euclid's Elements. Euclid provided a proof that there are infinitely many prime numbers. He showed that if every umber 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. Another Greek mathematician that contributed greatly to prime numbers was Eratosthenes. He discovered the Sieve of Eratosthenes. This is an algorithm for calculating prime numbers up to a limit. The process goes that if one were to list out all natural numbers starting with 2 to infinity and take out every second number after two, then continue the process, one would end up with a list of prime numbers. Later in the 17th century, the mathematician Pierre de Fermat contributed to the prime number theorem with his Little Theorem. The theorem states that if we have two numbers a and p, and p is a prime number, p−1 then ax p will always have the remainder of 1. This is a way for mathematicians to test out larger prime numbers. There are also a set of prime numbers known as Fermat primes that can be obtained from the formula 22n + 1 However, later mathematicians Euler discovered that 223 − 1 is a composite number and Fermat's theorem no longer applied to all prime numbers. Later enters French Mathematician Marin Mersenne. Mathematician of the 17th century, Mersenne had Mersenne primes named after his honor. Mathematicians, before Fermat, thought that the form of 2n − 1 would account for every prime number. Mersenne proved how this formula was correct only when n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. Therefore, every n < 257 would be composite., or not prime. Mersenne primes are therefore every prime number when 2n − 1. In the 19th century, Friedrich Bernhard Riemann gave a hypothesis that was later known as the Riemann Hypothesis. The Riemann Hypothesis is said to be one of the most important contributions to understanding prime numbers. It shows the distribution of prime numbers. The hypothesis has origins of when Riemann studied the zeta function, in which Euler gave. The Riemann zeta function, or the Euler-Riemann zeta function is an infinite series in which all prime numbers can be considered. 1 1 1 X = Y n3 1 − p−s n=1 p where the right hand side extends all prime numbers and the left hand side is an infinite sum. In the 19th century, German mathematician Carl Friedrich Gauss was the first to find a pattern with prime numbers at only 15 years old. He graphed prime numbers and found that if they increased by 10 then the probability of prime numbers reduces to about 2. His method was written into the formula n π(x) ∼ log n Gauss' method was not able to be proved by himself. However in 1838, Peter Gustav Lejeune Dirichlet was able to expand on Gauss' idea. In Dirichlet's theorem on arithmetic progressions, he states that when there are two positive coprime integers exist, there exists infinitely many primes. The theorem takes the form a + nd where a and d are coprimes and n is a postive integer number. Chebyshev, a Russian mathematician of the 19th century was close to proving the prime number theorem. Chebyshev's main contribution was determining the sum of prime numbers p that can be less than or equal to x. His function was very important in proving the prime number theorem. θ(x) = X log p p≤ 2 His second relation is also used extensively in proving the prime number theorem X (x) = log p pk≤x This function is used to calculate the sum of all powers of prime numbers of pk ≤ x In 1896, mathematicians Hadamard and de la Vall´eePoussin individually proved the prime number theorem. Hadamard proved the prime number theorem in 1896 with the statement that the number of primes ≤ n tends to 1 as n . de la Valle Poussin stated that π(x), the number of primes ≤ x, tends to x as x tends to ln(n) loge x infinity. These mathematicians used complex analysis of the Riemann hypothesis and with help of Chebyshev's function they were able to prove the prime number theorem. An 'elementary' proof for The Prime Number Theorem was yet to be awaited. Today, calculating primes has entered the computer age. In 1951, mathematicians Miller and Wheeler began to compute prime numbers using computers. They found several primes and a 79 digit record. 180(2127 − 1)2 + 1 A year later, Raphael Robinson discovered five new Mersenne primes using the Standards Western Automatic Computer. They were the largest prime numbers at the time of Robinson. Each consisted of 157, 183, 386, 664 and 687 digits. However, in 1948, Alan Turing was working on a machine that was able to find prime numbers. Sometimes in history, they attributed the findings of primes by a computer to Turing. In January of 1996, the Great Internet Mersenne Prime Search was created. This internet based search uses personal home computers in search of Mersenne Primes. Anyone who would like to help make mathematical progress could easily sign up for this program. Before the program, there were only 34 known Mersenne Primes. On January 25, 2013, they were able to discover up to the 48th prime. Selberg and Erd¨os In the spring of 1948, An Elementary Proof of The Prime Number Theorem was published by Atle Selberg. This proof, unlike Hadamard's and de la Vall´eePoussin's, uses no complex analysis of any kind. Now this paper is famous for many reasons. Not only is it the first elementary proof of The Prime Number Theorem, but also the history behind the mathematicians that made it possible. Atle Selberg was highly influenced by mathematics from a young age. His father was a highly renowned mathematics teacher but it was the many books on his fathers library that sparked Selberg's interest. Not only did Selberg enter the field of mathematics, three of his eight siblings were as well. Selberg, from an early age, was fascinated by real numbers. This was when Selberg's brother, Sigmund Selberg, introduced him to Chebyshev and his work amongst prime numbers. It was not until he came across the works of Ramanujan in the school library when he became interested in studying number theory. Everything about Ramanujan intrigued Selberg and pushed him into making his own mathematical work. He wrote his first 23 page paper, 'On Some Arithmetical Identities,' while still in high school. He went on to attend the University of Oslo where he obtained his doctorate. His thesis was 'On The Zeros of Riemann's Zeta-function.' He had some difficulties in his life due to World War 2. After he married and had two children, Selberg went onto Princeton to teach, where he met the mathematician Paul Erd¨os. Paul Erd¨oswas born druing a rough time for his parents. Just days before he was born, his two sisters passed away from scarlet fever. Consequently, this had made Erd¨osisolated and very well protected by his parents.
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