On 7 June 1742, Christian Goldbach wrote to Leonhard Euler saying (in more modern terms):
- Every integer greater than 5 can be written as the sum of three primes.
Euler noted that this is equivalent to:
- Every even integer greater than 2 can be written as the sum of two primes,
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:
- $\forall{\quad}n\geq{2},{\quad}\exists{\quad}p{}rimes{\quad}p,q{\quad}s.t.{\quad}p+q=2n.$
Less formally:
- For all (integers) n greater than or equal to two, there exist prime numbers, p and q such that $p+q=2n.$
Now let's try some examples:
- 4=2+2 (notice that the primes, p and q need not be distinct.)
- 6=3+3
- 8=3+5
- 10=3+7 (and also 5+5)
- 12=5+7
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 http://en.wikipedia.org/wiki/Goldbach's_conjecture
CategoryMaths
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