A prime number is a positive integer (whole number) which is only divisible by itself and one.  The first few are two, three, five, seven and 11.  An integer which is not prime is said to be composite.

The fundamental theorem of arithmetic states that every integer has a unique prime factorisation.  So, any composite number can be obtained by multiplying specific prime numbers.

Over 2,000 years ago, Euclid proved that the number of primes is infinite, in a beautiful proof by contradiction.  To understand this proof, consider the set containing the first three primes {2,3,5} and let q = (2×3×5) + 1 = 31.  This number is not divisible by either two, three or five. Now suppose the set of primes is the finite set {p_₁, p_₂, …, p__n} .  Consider q = (p_₁×p_₂×…×p__n ) + 1. The number q is not in the given set of primes, and it is not divisible by any of the primes in the set. Hence, it cannot be composite – a contradiction!  Therefore, there are infinitely many primes.

By looking at the first few primes, one can see that the gaps between primes vary.  A very interesting result shows that prime gaps can be arbitrarily long.  To show this, we use the factorial function n! which is the multiplication of all the integers from one to n.  For example, 6! = 6×5×4×3×2×1 = 720.

By looking at the first few primes, one can see that the gaps between primes vary

Consider the sequence of the five consecutive integers  6! + 2, 6! + 3, 6! + 4, 6! + 5, 6! + 6.  The first number (6×5×4×3×2×1) + 2 is divisible by two, the second number is divisible by three, the third by four and so on.  This sequence can be generalised to n! + 2, n! + 3, …, n! + n, giving a sequence of (n - 1) consecutive composite numbers, and hence an arbitrarily long prime gap.

The distribution of prime numbers seems random, and hence it appears difficult to establish a formula that generates them.  However, such formulae have been discovered.  The figure shows a formula that generates the n^th prime number.  It was published by C.P. Willans in 1964 and uses a theorem by John Wilson which states that a number x is prime precisely when the number (x - 1)! + 1 is divisible by x.  However, the required computational power is not available to produce beyond the first few primes.  To date, no formula exists that can be used to efficiently compute prime numbers.

Vincent A. Mercieca and Christina Zarb are mathematics lecturers at the University of Malta Junior College.

Sound Bites

•        Twin primes are two prime numbers which differ by two, such as three and five, or 29 and 31.  One of the most famous open questions in mathematics is the twin prime conjecture, which states that there are infinitely many twin primes.

•        In May 2013, Yitang Zhang showed that there are infinitely many pairs of primes which differ by 70 million.  Although far from the twin prime result, it was the first result of this kind.  Using a different technique, James Maynard reduced the gap to 600.  Since then, the polymath project, founded by Sir Timothy Gowers, to create a collaborative platform in solving difficult mathematical problems, has reduced this gap to 246.  The largest twin primes known to date are 2996863034895 ((2^^1290000) ) ± 1.

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DID YOU KNOW?

•        Large prime numbers are essential to modern cryptography.  The idea is that the product of two large prime numbers is easy to compute, but the reverse process of factorisation is very difficult.

•        A Mersenne prime is a prime of the form 2^ⁿ - 1, where 2^ⁿ  is the number two multiplied by itself n times.  They are named after Marin Mersenne, a 17th-century French friar.  The primes 3 = 2^² - 1 and 7 = 2^³ - 1 are examples. 

•        The Great Internet Mersenne Prime Search (GIMPS), founded in 1996 by George Woltman, is a collaborative project of volunteers who use freely available software to search for Mersenne primes.  To date, a total of 17 Mersenne primes have been found, 15 of which were the largest known primes at the time of their discovery.

•        The largest known prime to date is 2^^82,589,933  - 1, discovered in 2018 by Patrick Laroche.

For more trivia see: www.um.edu.mt/think.