The Prime Number Theorem: An Examination of Part 5 by Sebastian Augusthy 1) Background on the prime number theory A Prime number is any natural number that is divisible by itself and the number 1. For example 2; 3; 5; 7; 11; 13; 19 are just a few of the numbers. Prime numbers are useful because all integers can be written as products of 2 or more prime numbers. Ancient Greek mathematicians rst studied prime numbers and their properties extensively. There are no real patterns to prime numbers. Some unique prime numbers are ascending primes and palindromic primes. Ascending prime, for example, is 1234567801234567891. Palindromic primes are primes that are the same backwards and forward. For example, 111191111 and 919191919 are palindromic primes. The Fundamental Theorem of Arith- metic states that any natural number that is greater than one is the product of a unique set of prime numbers. The sequence of the numbers can be rear- ranged in the list, but an ordered set would be unique. Two examples that explain this concept are: 7 = 7, since 7 is prime. 84 = 2 ∗ 2 ∗ 3 ∗ 7 is the unique decomposition of 84 into primes, ordered by increasing value of the primes. One could also write 84 = (22) ∗ 3 ∗ 7. There are many types of prime numbers. Two main prime numbers are Mersenne prime numbers, which are in the format 2n − 1 for example, 22 − 1 = 3 and Fermat primes which are in the format 2(2n) + 1 For example 2(20) + 1 = 2 + 1 = 3 is a Fermat prime. There are many other kinds of prime numbers such as the Wieferich primes, Wall-Sun-Sun primes, Wilson primes and Wolstenholme primes. Mathematicians of Pythagora's school were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers. A perfect number is one whose proper divisors sum to the number itself. For example, the number 6 has proper divisors 1; 2 and 3 and 1 + 2 + 3 = 6; 28 has divisors 1; 2; 4; 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28 A pair of amicable numbers is a pair of numbers like 220 and 284 such that the proper divisors of one number sum to the other and vice versa. By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are innitely many prime numbers. This is one of the rst proofs known, which uses the method of contradiction to establish a result. Euclid also provided proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way. Euclid also showed that if the number 2n1 is prime, then the number 2n − 1(2n − 1) is a perfect number. The fact that prime numbers have no real pattern and there are many types of primes makes them really useful. One use of prime numbers familiar to many in the elds of cryptology and of computer security is to multiply a pair of large numbers for date encryption. Because it is dicult to determine factors of a large number, this is a fairly secure way of encrypting computer data. Cryptologists use prime numbers to come up with ways to break codes. Prime numbers are used to keep messages hidden. Banks use prime numbers to make computer keys for people. These keys are made by multiplying two prime numbers together to make a new number C. The bank keeps the prime numbers hidden for their use and gives the C number out to the public. Prime number encryption works as soon as a consumer inputs a credit card number online. The RSA (Rivest-Shamir-Adelman) algorithm uses a public key and a private key to hide information from possible thieves. The public key is available to the public, but it is hard to break because it is a product of two smaller prime numbers. The reason this RSA algorithm works is because the number used to encrypt the data with a public key is a very large number that is very hard to factor into prime numbers. The University of California at Berkeley states that it is simple to multiply two very large prime numbers together, but very dicult to break that gure down unless done with the help of a very powerful computer. The overall encryption program works using the example x=yz where x is the public key, while y and z are private keys. These private keys are very large prime numbers multiplied together, sometimes to the order of 100 or 200 numeric places. The two private keys unlock the public key and decode the information sent through the transmission. Whether faster algorithms or quantum computing will ever make this approach obsolete are questions that remain unsolved. Many mathematicians have made contributions to the eorts to understand prime numbers. To begin with, lets start with Gauss. Gauss was born on April 30, 1777 in Braunschweig, Germany. He was a child prodigy who was able to nd a discrepancy in his father's paycheck. Later on in Gauss's life he would do great things. One of the great things that Gauss did was to graph the density of prime numbers up to a certain number. He found out that the more prime numbers he placed on the graph, the more dense they spread of prime numbers were. This was important because he was able to show that prime numbers will always be increasing. This makes him a highly regarded German mathematician whose works included contributions to the number theory, geometry, probability theory, geodesy, planetary astronomy, the theory of functions and potential theory (including electromagnetism). Gauss proved 1 that π(x) ≈ li(x) (the principal value of integral of for u from u = 0 to log u = x). Another mathematician who studied prime numbers was Chebyshev. He was born on May 16, 1821 in Okatovo, Russia. Chebyshev became assistant professor of mathematics at the University of St. Petersburg. He is remem- bered primarily for his work on the theory of prime numbers. Chebyshev proved one of Joseph Bertrand's conjectures which state that for any n > 3 there must be a prime between n and 2n. This was important because he said that between two numbers there must be primes. In 1845 Bertrand conjec- tured that there was always at least one prime between nand 2n for n > 3. While proving Bertrand's conjecture in 1850, Chebyshev also came close to (π(n) log n) proving the Prime Number Theorem, proving that if (with π(n) n the number of primes ≤ n had a limit as n ! 1 then that limit is 1. The proof of this result was only completed two years after Chebyshev's death. Euler was another mathematician who studied prime numbers. Euler was a Swiss mathematician who was famous for his work in innitesimal calculus and graph theory. During his lifetime he worked on prime numbers. In 1772, Euler noticed that p(n) = n2 + n + 41 was prime for all natural numbers less than 40. This was a great formula to have because it was another formula that was able to gure out prime numbers in a given set. He was a Swiss mathematician and physicist, and one of the founders of pure mathematics. He not only made decisive and formative con- tributions to the subjects of geometry, calculus, mechanics, and number theory but also developed methods for solving problems in observational astronomy and demonstrated useful applications of mathematics in technology and public aairs. Similarly, another similar mathematician was Fermat. He was born on August 17, 1601 or 1607 (there is a dispute as to the year of his birth). He is known for his work in analytic geometry, probability, and optics. While working on prime numbers, Fermat came up with the formula 22n + 1 in order to nd prime numbers. This was an important formula because it helped in nding more prime numbers. Fermat is known to have collaborated with various mathematicians to decipher many concepts and theories. Along with Rene Descartes, Fermat was considered one of the two leading mathe- maticians of the rst half of the 17th century. They collaborated on many mathematical ideas, which continue to be used in today's world. Indepen- dently of Descartes, Fermat discovered the fundamental principle of analytic geometry. His methods for nding tangents to curves and their maximum and minimum points led him to be regarded as the inventor of the dierential cal- culus. His relationship with Blaise Pascal he became the co-founder of the theory of probability. One of the most elegant of these had been the theorem that every prime of the form 4n + 1 is uniquely expressible as the sum of two squares. A more important result, now known as Fermat's lesser theorem, asserts that if p is a prime number and if a is any positive integer, then ap - a is divisible by p. Fermat rarely gave demonstrations of his results, and in this case proofs were provided by Gottfried Leibniz, the 17th-century Ger- man mathematician and philosopher, and Leonhard Euler, the 18th-century Swiss mathematician. For occasional demonstrations of his theorems Fermat used a device that he called his method of innite descent, an inverted form of reasoning by recurrence or mathematical induction. Another mathematician who studied prime numbers was Marin Mersenne. He created his own way of nding primes called Mersenne primes. Mersenne primes were prime numbers calculated using the formula 2n − 1 He is credited with nding yet another way to nd prime numbers.