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more and more likely to be true the bigger the even? Primes become more rare, so it seems to me this notion is counter intuitive. A few recent papers all point to that Goldbach becomes more and more likely the higher up you go.

A very large even can be the sum of two large odds or 1 small odd and 1 very large odd. The first case, at least by intuition, should eventually fail for a very large even, since both large odds would increase and thus decrease both their chances of being prime. In the second case, suppose a small odd is kept fixed at a prime, the very big odd would have to increase. For instance, if you fix one of the odds at 3, the other odd must increase as the even increases, which decreases the chance of the other odd of being a prime. Again, at least by intuition, the second case should eventually fail for a very large even.

The average gap between two consecutive primes grows very slowly. As I saw from the data I worked with, by the time the first 240 million primes were reached, the average gap between two consecutive primes only grows to about 20 to 30 or so, even though gaps of over 300 between consecutive primes have already occurred. Perhaps, in my opinion, when an even surpasses 10 to the power of 1000, one cannot find two primes to add to that even.

A very large even can be the sum of two large odds or 1 small odd and 1 very large odd. The first case, at least by intuition, should eventually fail for a very large even, since both large odds would increase and thus decrease both their chances of being prime. In the second case, suppose a small odd is kept fixed at a prime, the very big odd would have to increase. For instance, if you fix one of the odds at 3, the other odd must increase as the even increases, which decreases the chance of the other odd of being a prime. Again, at least by intuition, the second case should eventually fail for a very large even.

The average gap between two consecutive primes grows very slowly. As I saw from the data I worked with, by the time the first 240 million primes were reached, the average gap between two consecutive primes only grows to about 20 to 30 or so, even though gaps of over 300 between consecutive primes have already occurred. Perhaps, in my opinion, when an even surpasses 10 to the power of 1000, one cannot find two primes to add to that even.

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