The lack of a pattern in prime numbers is something that has perplexed and frustrated mathematicians for centuries. This is because most things in life can be explained by mathematical patterns, such as the Fibonacci sequence, also known as “nature’s numbering system”. In my EPQ, I am going to explore if there is a pattern in prime numbers and, if so, if there is a means of predicting them that is simpler than the one that we use today. Currently, to prove a number is prime it has to be divided, in turn, by all the prime numbers smaller than it. It has to result in a remainder in every case to be classified as prime. If it is divisible by even one other number that isn’t itself or one, it is no longer classified as prime. This means that thousands of calculations can be done before finding out that a number suspected to be prime (known as a pseudo prime) is not actually prime. This is a time consuming method to use for larger numbers that may need to be divided thousands of times before being classified as prime and can be frustrating and yield little result. Mathematicians, to this day, have struggled to understand why prime numbers occur as randomly as they do. I will now explore this area of mathematics and try to come to a conclusion as to whether we can predict the occurrence of prime numbers.
A prime number is defined as a natural number greater than one which can only be divided by one and itself, with no remainder. In other words, it can not be written as a product of two natural numbers that are smaller than itself. The lowest prime number is the number two, which is also coincidentally the only even prime number. One is not considered to be a prime number as it is only divisible by one number (one) and is not greater than the number one, so falls short of our definition of a prime number. All prime numbers greater than 5 must end in a 1, 3, 7 or a 9 as it is impossible for a prime to end in an even number (as it will be divisible by 2) or a 5 (as it will be divisible by 5).To prove whether or not a number is prime, you try to divide it by the lowest prime number, two. If this forms a whole number, then the number being tested is not prime but is in fact a composite number (numbers with more than two factors, that are made when more than two primes are multiplied together). If it does not form a whole number you continue by dividing it by the next prime number till you get a whole number or exceed the value of the number that you are testing. This method is known as trial division and is a simple way to determine whether or not a small number is prime. As previously mentioned this method is not suitable for larger numbers, so, instead algorithms such as the AKS Primality Test and Miller-Rabin Primality Test are used.
Prime numbers are considered to be the ‘building blocks’ of mathematics as every number in existence is either a prime number or can be made by multiplying prime numbers together. Prime numbers are a major component of number theory (also known as arithmetic), a branch of pure mathematics that studies integers. For years, mathematicians have looked into this area of mathematics but have never been able to construct a formula for prime numbers. This is because prime numbers go on indefinitely and there is no largest prime number, as proved by Euclid in 300 BC. As numbers get larger and larger, the frequency of the occurrence of prime numbers becomes less as there are more possible factors for the number to be divided by. At the time of writing, the largest known prime number is 277,232,917−1 and was found in 2017 by the Great Internet Mersenne Prime Search (GIMPS). It has 23,249,425 digits and is most commonly known as M77232917. The search for numbers that are both prime and larger than M77232917 is ongoing and it is unknown when the next largest prime number is likely to be found and how many digits it’s likely to possess. The process is long and arduous and it could be sped up considerably if a formula was available for dictating prime numbers. In recent years, mathematicians have made massive steps forward in this area, leading some to believe that a formula/pattern in prime numbers has now been found.
History of Primes
The study of prime numbers dates back to the ancient Greek mathematicians. The idea of primality was first considered in Pythagoras’ schools in Ancient Greece between 500 and 300 BC. Although the study in Pythagorean Schools, was more concerned with understanding perfect and amicable numbers (whose links to prime numbers I will discuss later). The earliest records of the concentrated study of prime numbers date back to 300 BC in a book published by Euclid called Elements. Euclid (also known as Euclid of Alexandria) is a Greek mathematician who is widely known for his work in geometry. Euclidean geometry is the study of solid figures and planes on the basis of Euclid’s theorems which are still being used in the teaching of geometry today. We have him to thank for creating the first definitions for many terms used in geometry such as angle (“the inclination of two lines”), circle, line (“a length without breadth”) and triangle. He produced a book called Elements which is widely regarded to be the most influential book in the history of mathematics. This book contains many theorems deduced by Euclid in the area of euclidean geometry alongside work in the areas of number theory and spherical geometry. Whilst Euclid is best known for his work in geometry, he helped the current understanding of prime numbers to progress immeasurably. In Book IX of Elements, Euclid proves that prime numbers go on indefinitely, something that still remains true to this day. To prove this to be the case, he used the method of proof by contradiction, one of the first proofs known to be correct. This proof is shown in figure 1 in the appendix.
Euclid can also be credited with stating that “if 2n – 1 is prime then the number 2n-1(2n – 1) is a perfect number”. In number theory, a perfect number is defined as a positive integer that is equal to the sum of its positive proper divisors. This theorem is explained by Euclid in Elements where he states “if a finite geometric series beginning at 1 with ratio 2 has a prime sum P, then this sum multiplied by the last term T in the series is perfect.”. This theorem is now referred to as the Euclid-Euler Theorem as Leonhard Euler, a Swiss mathematician (around much later in 1700’s), was able to successfully prove that all even perfect numbers can be calculated using Euclid’s statement. It is unknown to this day whether or not there are any odd numbered perfect numbers. The next advancement in our understanding/computation of prime numbers is almost 100 years later when another Greek mathematician Eratosthenes created the first known algorithm for computing prime numbers, known as the Sieve of Eratosthenes. It is a simple algorithm used to compute all prime numbers up to a certain limit. It works by marking multiples of prime numbers as composite, starting with the lowest prime 2, till the only numbers that are left are prime.
There is then a long duration of time where either there is little known about the advancements made concerning prime numbers or little advancements themselves were made. The next major development occurred in the 1600’s by a French lawyer/mathematician, called Pierre de Fermat who is most widely remembered for creating a method of finding the smallest and greatest ordinates of curved lines. He is also known for studying and defining a group of numbers called Fermat numbers, whilst also noting their significance. These are a positive integer in the form 22n – 1 (2 to the nth power of 2) where n is a nonnegative integer. A Fermat Prime is said to be a Fermat number that is also prime. Pierre de Fermat believed that all Fermat numbers would be prime, something that has since been proved to be incorrect. This is the case when n is set equal to 5 as the resulting number 4294967297 is divisible by 641 so can not be prime. As of today there are only 5 known Fermat Prime Numbers where the value of n equals either 0, 1, 2, 3 and 4. This is a far cry from Fermat’s original deduction that all Fermat numbers would be prime as that is only true up until the number 4. However, the search for Fermat numbers where n is greater than 4 is still ongoing.
Pierre de Fermat is also well known for creating many theorems including one that regards prime numbers. Fermat’s Little Theorem states that “if p is a prime number, then for any integer, a, the number ap−a is an integer multiple of p”. This theorem is used in many different primality tests including Fermat’s Primality Test and the Miller-Rabin Primality Test. The theorem is also used on today’s electronic computers as part of the calculations. It is also one of the fundamental theorems used in number theory. Once more, this theorem was then first proved to be true in a publication by Euler in 1736. Although there is some dispute over whether he was actually the first one to prove it to be true as a German philosopher in the history of mathematics called Gottfried Leibniz gave the same proof in an unpublished document in 1683. Euler’s Theorem is said to be a generalisation of Fermat’s Little Theorem. Fermat’s Little Theorem is also related to many other theorems in number theory including Carmichael’s Theorem and Lagrange’s group theory Theorem.
Marin Mersenne was a French monk and polymath who was alive in the 1600’s. Unlike others, his work was not concentrated in one area as he researched mathematics and science whilst also lecturing in philosophy and writing about musical theory. He is best known in the mathematical world for his work studying and researching prime numbers and Number Theory. He was working in this area at the same time as Fermat and the two mathematicians were frequently in contact, in a time were there were no scientific journals or forums. They worked together, at times, in their pursuit of greater understanding of prime numbers. Mersenne specifically, extensively, studied numbers of the form 2n – 1 after Fermat spoke about numbers of the form 2n +1. Not all numbers in this form are prime but any number that is of this form, where n is a prime number, that turns out to be prime is called a Mersenne Prime number in recognition of his work. Finding Mersenne Primes, like finding most primes, is laborious and takes a long time. In 1996 the GIMPS (Great Internet Mersenne Prime Search) started searching for Mersenne Primes using processing power on computers to help with the amount of mathematical computation necessary. This level of computer power was not available before this time. Currently, 450 trillion calculations can be done each second. This may sound like a lot of calculations, but it has only resulted in the finding of 50 Mersenne Primes to date (2018) as the amount of calculations needing to be done to prove primality is so large.
As already mentioned, Leonhard Euler has had, arguably, the biggest impact on prime numbers and number theory as a whole. He helped to prove that many theorems, such as Newton’s identities and Fermat’s Little Theorem (which has now been generalised to Euler’s Theorem), expressed by previous mathematicians were correct whilst also completing his own work in the sector. Leonhard Euler was a Swiss mathematician that studied in almost all areas of mathematics in the 1700’s and is considered, by some, to be one of the greatest mathematicians of all time. He largely contributed towards our understanding of analytic number theory whilst also working in a variety of other areas including, but not limited to, mechanics, optics and fluid dynamics. A highlight in his mathematical career would be having two numbers named after him, e (known as Euler’s number) and γ (known as the Euler-Mascheroni constant).
A lot of his work in the area of number theory and prime numbers was based on the work of Pierre de Fermat and inspired by Christian Goldbach. As previously mentioned he worked on disproving and developing the work set out by Fermat. He discovered the connection between the Riemann Zeta Function and prime numbers. It is of great significance in the area of number theory as it is used in relation to prime numbers. Euler proved that the Zeta function can be written as a product of prime numbers, as shown in figure 3 in the appendix. This is known as the Euler product formula for the Riemann Zeta Function which is also used in physics. Euler is also known for developing the basis of what is now known as the law of quadratic reciprocity. This is one of the few occasions in his career when he discovered something but was not able to prove it. It is a theorem that gives the conditions for solving quadratic equations modulo prime numbers. It is shown in figure 4 in the appendix. Euler also had a great interest in perfect numbers and proved a relationship between perfect numbers and Mersenne Primes. This work was the basis of the work in this sector done by Carl Friedrich Gauss. One of Euler’s last contributions, in 1772, to number theory was the discovery of a Mersenne prime equivalent to 231 – 1 (2147483647) which remained to be the largest known prime number for over 90 years.
Carl Friedrich Gauss, a German mathematician in the 1800’s, was one of the first mathematicians to look into the distribution of prime numbers. At first it appears that they are distributed randomly as one group of 100 numbers contains 9 (the 10 before 10000000) whereas another contains only 2 (the 10 after 10000000). In his lifetime Gauss did many calculations concerning the density of prime numbers, it is believed that Gauss counted all the prime numbers up to 3 million during his lifetime, an achievement not broken by any other mathematician without access to a computer. He came to the conclusion that “for large n, the density of primes near n is about 1/log(n). This statement formed the basis of prime number theorem, something that I will explore later in more detail. Over the years various mathematicians attempted to prove his statement but it wasn’t until 1896 until the result was proved, using complex analysis, by Hadamard and de la Vallée Poussin