Goldbach's Conjecture

On 7 June 1742, Christian Goldbach wrote to Leonhard Euler saying (in more modern terms):

Euler noted that this is equivalent to: He then added that he regarded this an entirely certain theorem, in spite of his being unable to prove it.

This remains an unsolved problem.

In notation

Using standard mathematical notation we can write this as follows:

Less formally: Now let's try some examples:

We could carry on but no amount of examples will ever prove the statement true "for all even integers greater than 2". Of course, if we found an even integer greater than 2 which is not the sum of two primes then this would be enough to prove Goldbach's Conjecture false.

This is a celebrated conjecture which, not surprisingly, has been checked by computer for large numbers (up to $10^{14}).$

For more details see's_conjecture

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