One of the oldest and most famous unsolved mathematical problems is the Goldbach conjecture. This is
Every even number greater than 2 can be expressed as the sum of two prime numbers.
This problem was first posed in 1742 by the German mathematician Christian Goldbach and nearly three hundred years later no one has managed to prove whether or not it is true.
A prime number (or prime) is a positive number, which only has factors of itself and 1. For example: 2, 3, 5, 7, 11, 13, 17, 19 etc. By convention 1 is a not considered to be a prime number. Clearly all prime numbers other than 2 must be odd.
I’ve illustrated the Goldbach conjecture for some even numbers below:
4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 5 + 5 OR 3 + 7
100 = 3 + 97 OR 11 + 89 OR 17 + 83 OR 29 + 71 OR 41+ 59 OR 47 +53
In general, the larger the even number the more different ways it can be split between two primes.
Proving the Goldbach Conjecture
Shortly after the Goldbach stated the conjecture one of the greatest eighteenth century mathematicians Leonhard Euler said:
‘That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it’
Leonard Euler one of the greatest ever mathematicians – Image from Wikimedia Commons
For the last 300 years mathematicians have tried without success to prove it, or to disprove it by finding an even number which cannot not be expressed a sum of two primes. Before the advent of computers, to try to disprove the conjecture, mathematicians had to laboriously check even numbers by hand to look for a number which could not be expressed as the sum of two primes. In 1938 the mathematician Nils Popping showed that all even numbers up to 100,000 could be expressed as the sum of two primes. This must have been an incredible tedious task.
Today, using super computers the Goldbach conjecture has been verified to be true for all even numbers less than four million trillion (4 000 000 000 000 000 000). However, this impressive feat does not prove the conjecture is true for all numbers. All it shows is that the conjecture is true for all even numbers less than four million trillion. It is possible that there exists an even larger number which cannot be split into two primes. What has never been found is a mathematical proof that the conjecture is true for all even numbers
The Bloomberg Prize
In a 1992 novel Uncle Petros and Goldbach’s Conjecture by Apostolos Doxiadis the anonymous narrator describes his fascination with his reclusive Uncle Petros, who is considered a failure by his family. When his nephew shows an interest in mathematics, Petros offers him a problem to solve
prove that any even number greater than 2 is the sum of two primes.
The narrator works hard on trying crack this problem all summer without success. He later learns that it is more than 250 years old and has remained unsolved by the greatest mathematicians.
Enraged and frustrated, he confronts his uncle, only to discover that Petros has been psychologically crippled by the Goldbach conjecture for decades. When he was a young scholar determined to pursue distinction in the world of mathematics, Petros decided to tackle the proof. But as the proof revealed itself to be beyond him, his pursuit became a nightmare in which he imagined that numbers had taken human form and were speaking to him. As Petros lost hope of finding the proof, he lost his grip on his sanity and his livelihood as a professor at the University of Munich.
In March 2000 the publishers of the book (Bloomberg in the USA and Faber and Faber in the UK) offered a prize of one million dollars anyone who could prove the Goldbach conjecture and whose proof was accepted by fellow mathematicians. The prize was kept open for two years but nobody claimed it.
Other unsolved problems
In the year 2000 the Clay Mathematics Institute produced a list of of seven mathematical problems for which they would give a million dollar prize for the person(s) who solved them The Goldbach Conjecture was not among them. I have listed the seven problems below and I think that most of my readers will agree that they cannot be understood by anyone without a high-level knowledge of mathematics. To me the great attraction of the Goldbach conjecture is its simplicity to state and its impossibility to prove!
For another unsolved (but less famous) problem see https://explainingscience.org/2017/08/12/196-an-unsolved-problem/