Alex's Adventures in Numberland - Alex Bellos [107]
On a lighter note, here’s a nursery sequence:
(A38674) 2, 2, 4, 4, 2, 6, 6, 2, 8, 8, 16
These are the numbers from the Latin American children’s song ‘La Farolera’: ‘Dos y dos son quatro, cuatro y dos son seis. Seis y dos son ocho, y ocho dieciseis.’
But perhaps the most classic sequence of all is the prime numbers:
(A40) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…
Prime numbers are the natural numbers greater than 1 that are divisible only by themselves and 1. They are simple to describe but the sequence exhibits some rather spectacular, and sometimes mysterious, qualities. First, as Euclid proved, there is an infinite number of them. Think of a number, any number, and you will always be able to find a prime number higher than that number. Second, every natural number above 1 can be written as a unique product of imes. In other words, every number is equal to a unique set of prime numbers multiplied by each other. For example, 221 is 13×17. The next number, 222, is 2×3×37. The one after that, 223, is prime, so produced only by 223×1, and 224 is 2×2×2×2 × 2 × 7. We could carry on for ever and each number could be winnowed down to a product of primes in only one possible way. For example, a billion is 2×2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5. This characteristic of numbers is known as the fundamental theorem of arithmetic, and is why primes are considered the indivisible building blocks of the natural number system.
Primes are also building blocks when we add them together. Every even number bigger than 2 is the sum of two primes:
4 = 2 + 2
6 = 3 + 3
8 = 5 + 3
10 = 5 + 5
12 = 5 + 7
…
222 = 199 + 23
224 = 211 + 13
…
This proposition, that every even number is the sum of two primes, is known as the Goldbach Conjecture, named after the Prussian mathematician Christian Goldbach, who corresponded with Leonhard Euler about it. Euler was ‘entirely certain’ that the conjecture was true. In almost 300 years of looking and trying, no one has found an even number that is not the sum of two primes, but so far no one has actually been able to prove that the conjecture is true. It is one of the oldest and most famous unsolved problems in mathematics. In 2000, so confident were they that a proof was still beyond the limits of mathematical knowledge that the publishers of the mathematical detective story Uncle Petros and Goldbach’s Conjecture offered a $1,000,000 prize for anyone who could solve it. No one did.
The Goldbach Conjecture is not the only unresolved issue regarding the primes. Another focus of study is how they seem to be scattered unpredictably along the number line, with no obvious pattern to the sequence. In fact, the search for the harmonies that underpin the distribution of the primes is one of the richest areas of enquiry in number theory, and it has led to many deep results and suppositions.
For all their pre-eminence, however, the primes do not have exclusive claim among the sequences to holding special secrets of mathematical order (or disorder). All sequences contribute in some way to a greater appreciation of how numbers behave. Sloane’s On-Line Encyclopedia of Integer Sequences can also be considered a compendium of patterns, a Domesday Book of mathematical DNA, a directory of the underlying numerical order of the world. It might have sprung from Neil Sloane’s personal obsession, but the project has become a truly important scientific resource.
Sloane compares the Encyclopedia to a maths equivalent of the FBI fingerprint database. ‘When you go to a crime scene and you take a fingerprint, you then check it against the file of fingerprints to identify the suspect,’ he said. ‘It’s the same thing with the Encyclopedia. Mathematicians will come up with a sequence of numbers that occurs naturally in their work, and then they look it up in the database – and it’s lovely for them if they find it there already.’ The database’s usefulness is not restricted to pure mathematics.