The conjecture that every even number may be expressed as the sum of 2 prime numbers. This conjecture has yet to be proved or disproved. The conjecture has been tested numerically up to huge limits (4 × 1011?), and a somewhat weaker theorem has been proved: that every even number is the sum of not more than 300000 prime numbers.
None of this provides any indication as to whether the conjecture is, in fact, true. Indeed, it might be that the conjecture is unproveable in Peano arithmetic. But this would have interesting consequences: suppose the conjecture is unproveable. Then it must be true in True arithmetic. For suppose it were false. Then a counterexample could be found, i.e. some (presumably huge, but finite) even number E would exist, that wasn't the sum of 2 primes. But finitely many tests would confirm E was a counterexample: just list all primes smaller than E, and show that the sum of every two of them is not E. So such a counterexample cannot exist, hence if Goldbach's conjecture is unproveable it must be true.