chapter  14
8 Pages

How can Anything so Simple be so Difficult?

WithMalcolm E Lines

Before leaving the counting numbers some reference must now be made to what (apart perhaps from Fermat’s Last Theorem) is the most famous of all unsolved problems concerning the integers. It is unusually frustrating because this particular hypothesis can be stated in terms which are almost unbelievably simple. First mentioned in an exchange of letters between Euler and the German mathematician Christian Goldbach in the year 1742, it is the suggestion that every even number can be written as the sum of two odd prime numbers. If the integer 1 is not considered a prime (which usually it is not) then the numbers 2 and 4 should be excluded, so that a more correct statement of the conjecture is that all even numbers larger than 4 can be expressed (usually in many different ways) as the sum of two primes.