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The modern version of Goldbach's Conjecture (called Goldbach's Strong Conjecture) is this:
Every even number greater than 2 is the sum of two primes.
Let's try a few:
4=2+2
6=3+3
8=3+5
10=3+7, 5+5
12=5+7
14=3+11, 7+7
16=3+13, 5+11
18=5+13, 7+11
20=3+17, 7+13
22=3+19, 5+17, 11+11
24=5+19, 7+17, 11+13
26=3+23, 7+19, 13+13
28=5+23, 11+17
30=7+23, 11+19, 13+17
32=3+29, 13+19
34=3+31, 5+29, 11+23, 17+17
36=5+31, 7+29, 13+23, 17+19
38=7+31, 19+19
40=3+37, 11+29, 17+23
The conjecture is looking safe so far. Not only is each even number the sum of two primes, but the number of pairs of primes tends to increase. This trend seems to continue. But no one has ever proved that this goes on forever. All of the even number up to 400,000,000,000 have been tested, so far, with no exceptions found.
Mathematicians have achieved some results in their efforts to prove (or disprove) this conjecture. In 1966, J. R. Chen showed that every sufficiently large even number is either the sum of two primes or of a prime and a near prime. A near prime is a number that is the product of two primes, like 91=7x13 or 4=2x2. No one knows just how large "sufficiently large" is.
There is another Goldbach Conjecture, that every odd number greater than 5 is the sum of three primes. This is known as the Weak Goldbach Conjecture. This too has not been proved or disproved. It has been shown that if there are exceptions, then there are only a finite number of exceptions. A slightly different form of these conjectures was originally posed by Christian Goldbach, in 1742.
Incidentally, if either Goldbach Conjecture is ever proven, then that would also prove that there are infinitely many primes. But we already knew that. See The Infinitude Of Primes. Unfortunately the fact that there are infinitely many primes does not imply either Goldbach Conjecture.