Every even number can be expressed as the sum of two primes.
One of the oldest unproven conjectures stems from a letter of 1742 from Goldbach to Euler. I was intrigued, for in seventy years I had not come across this type of concept, one which (as Euler declared) is certainly true even though it cannot be proven to be true.
I very soon convinced myself that the conjecture is sound, by an argument that many others have discovered and explored. The gap (g) between two adjacent prime numbers (p1, p2) grows as p1 increases, but grows far more slowly than p1 grows. For example the gap between the adjacent primes 977, 983, 991, 997 are (respectively) only 6, 7 and 6. As all primes (other that 2) are odd, any two added together will produce an even sum. It is easy enough to show for small even numbers (2n, 2(n+1), etc) that the conjecture holds: 4=3+1; 6=5+1, or 3+3; 8=7+1, or 5+3; 18=17+1, or 13+5, or 11+7. Thus we see that as 2n increases the number of ways in which it can be expressed as the sum of 2 primes also increases. So, even though it is a laborious task for computer enthusiasts to find the necessary primes, it becomes progressively harder to imagine the falsification of the conjecture.
But that is not a proof that the conjecture must, for all values of n, be true a priori. It is at least in part an induction a posteriori.