9. Aftermath

The Aftermath returns to some of the problems posed early in the book. The questions solved include: In which years does February have four Sundays? How many shuffles are needed to restore the order of the cards in a pack with two Jokers? How do prime numbers keep our credit cards secure? Are there infinitely many primes with final digit 9? Integers, squares and cubes, prime numbers and perfect numbers are revisited, with instructions for solving some of the ancient puzzles that inspired generations of mathematicians, and concluding with some of the unsolved problems in this exciting field of modern mathematics.

6. From cards to cryptography

‘From cards to cryptography’ applies another result of Fermat – his ‘little theorem’ – to the problem of finding the number of different coloured necklaces with a given number of beads and available colours, if we use at least two colours? Euler generalized this theorem, using his so-called ‘totient function’. Multiplying two prime numbers is relatively simple, but factorizing a large number into prime factors can be very difficult. This asymmetric process led to a method for encrypting messages, discovered independently by a former Bletchley Park codebreaker and by three mathematicians with the initials R, S, and A, hence the term ‘RSA encryption’.

4. Congruences, clocks, and calendars

‘Congruences, clocks, and calendars’ demonstrates how we might apply the idea of congruence, first introduced by Gauss in 1801, to problems such as testing which Mersenne numbers are primes and finding the day of the week on which a given date falls. Ancient Chinese puzzles depended on the solving of simultaneous linear congruences, inspiring mathematicians and giving rise to the Chinese Remainder Theorem. Exploring quadratic congruences leads towards the law of quadratic reciprocity, noted by Euler and Legendre and proved by Gauss. The problem, ‘Is 1066 a square or a non-square?’ can be solved by applying this law several times to reduce the numbers involved.

3. Prime-time mathematics

‘Prime-time mathematics’ explores prime numbers, which lie at the heart of number theory. Some primes cluster together and some are widely spread, while primes go on forever. The Sieve of Eratosthenes (3rd century BC) is an ancient method for identifying primes by iteratively marking the multiples of each prime as not prime. Every integer greater than 1 is either a prime number or can be written as a product of primes. Mersenne primes, named after French friar Marin de Mersenne, are prime numbers that are one less than a power of 2. Pierre de Fermat and Leonhard Euler were also prime number enthusiasts. The five Fermat primes are used in a problem from geometry.

8. How to win a million dollars

What is the Riemann hypothesis, and why does it matter? ‘How to win a million dollars’ looks in detail at Riemann’s conjecture. While Gauss attempted to explain why primes thin out, Bernhard Riemann in 1859 proposed an exact formula for the distribution of primes, employing Euler’s ‘zeta function’ and the idea of complex numbers. In 2000, the Clay Mathematics Institute offered a million dollars for the solutions of each of seven famous problems, of which the Riemann hypothesis was one. The Riemann hypothesis implies strong bounds on the growth of other arithmetic functions, in addition to the primes-counting function. It remains one of the most famous unsolved problems of mathematics.

7. Conjectures and theorems

‘Conjectures and theorems’ investigates a number of topics, such as the distribution of prime numbers, and two unsolved problems, Goldbach’s conjecture and the twin prime conjecture. The factorization of positive integers into primes is unique, but this does not hold for certain other systems of numbers. A more in-depth look at unique factorization gives deeper results, including a proposed result of Gauss. Mathematicians in the 1950s and 1960s confirmed that he was correct, as shown in the so-called ‘Baker-Heegner-Stark theorem’.

5. More triangles and squares

‘More triangles and squares’ explores Diophantine equations, named after the mathematician Diophantus of Alexandria. These are equations requiring whole number solutions. Which numbers can be written as the sum of two perfect squares? Joseph-Louis Lagrange’s theorem guarantees that every number can be written as the sum of four squares, and Edward Waring correctly suggested that there are similar results for higher powers. In 1637, Fermat conjectured that no three positive integers, a, b, and c, can satisfy the equation an+bn=cn, if n is greater than 2. Known as ‘Fermat’s last theorem’, this conjecture was eventually proved by Andrew Wiles in 1995.

2. Multiplying and dividing

‘Multiplying and dividing’ looks at multiples and divisors, focusing on the least common multiple and greatest common divisor of two numbers. We use Euclid’s algorithm as a method for computing the greatest common divisor of two numbers by using the division rule repeatedly. Perfect squares (integers that are the product of two equal integers) feature throughout number theory. Tests are given for divisibility by certain small numbers. An ancient method called ‘casting out nines’, was developed in India in around the year 1000, based on the argument that a number and its digital sum leave the same remainder when divided by 9. We can still use this method to verify the accuracy (or otherwise) of arithmetical calculations.

1. What is number theory?

‘What is number theory?’ puts number theory in its historical context, from the Pythagoreans to the present, explaining integers (whole numbers), prime numbers (the building blocks of number theory) squares and cubes, and perfect numbers (numbers whose factors add up to the number itself). How long can gaps between prime numbers be? Is there a formula for producing perfect numbers? Which primes can be expressed as a sum of squares? Other questions arise when we start adding primes.