To a large extent the primes appear to be distributed in a manner indistinguishable from randomly. There seems underneath to be no reason to believe that there are results like this. If they were random then there might be numbers that cannot be expressed as the sum of 3 primes, but somehow every number ends up so expressible.
I see this as backwards. Based on the prime number distribution we can actually predict the probability that a given number will be a counter example. And the sum over all positive integers of those probabilities is a miniscule number, so we're quite certain that the statement is true. We just don't know why.
The primes are distributed to a first approximation randomly with independent probabilities 1/log(n) for n being prime. You can refine this with obvious divisibility criteria. (For example odd numbers appear randomly prime with probability 2/log(n). And so on.)
If you first refine these statements for all the small primes that you care to, then make any prediction that you like, that prediction tends to hold up very well.
For example consider the even form of Goldbach's conjecture, which says that every even greater than 2 is the sum of 2 primes. Well, 4 is a special case, so it really is that every even greater than 4 is the sum of 2 odd primes. Does this seem unlikely?
If n is even, it is the sum of 2 smaller odd numbers in n/4 + O(1) ways. On average those have independent probabilities around 4/log(n/2)^2 of being the sum of 2 primes. Therefore it tends to be the sum of 2 primes in O(4n/log(n)2) ways, with the exact distribution being close to a Poisson distribution around the projected number, which for large n is well approximated by a normal distribution. (There are various minor quibbles and corrections that you can easily use to refine this basic prediction.) From this the probability of being a sum of two odd primes in 0 ways can be estimated, and turns out to go to 0 fast enough that we expect only a finite number of counter-examples. (2 and 4 are indeed counterexamples to the stronger statement, but we know of no others.)
This basic projection can be verified for small n by computer analysis, and indeed works out to be remarkably accurate. Therefore we have great confidence that Goldbach's conjecture is likely to be true, but absolutely no idea why.
The hope, some day, is to be able to close this gap. To not just make assertions based on known statistical facts about the distribution of primes, but to be able to prove them.
The best known example of a statement which should be true that we have not yet proven is the Riemann Hypothesis. It should be noted that Louis de Branges does claim to have proven it, nobody has investigated his claims depth, and he was right about the Bieberbach conjecture.
My favorite example of a statement which looks reasonable on the numerical evidence, cannot be supported by this kind of argument, which later proved false is Mertens conjecture.
I see this as backwards. Based on the prime number distribution we can actually predict the probability that a given number will be a counter example. And the sum over all positive integers of those probabilities is a miniscule number, so we're quite certain that the statement is true. We just don't know why.
The primes are distributed to a first approximation randomly with independent probabilities 1/log(n) for n being prime. You can refine this with obvious divisibility criteria. (For example odd numbers appear randomly prime with probability 2/log(n). And so on.)
If you first refine these statements for all the small primes that you care to, then make any prediction that you like, that prediction tends to hold up very well.
For example consider the even form of Goldbach's conjecture, which says that every even greater than 2 is the sum of 2 primes. Well, 4 is a special case, so it really is that every even greater than 4 is the sum of 2 odd primes. Does this seem unlikely?
If n is even, it is the sum of 2 smaller odd numbers in n/4 + O(1) ways. On average those have independent probabilities around 4/log(n/2)^2 of being the sum of 2 primes. Therefore it tends to be the sum of 2 primes in O(4n/log(n)2) ways, with the exact distribution being close to a Poisson distribution around the projected number, which for large n is well approximated by a normal distribution. (There are various minor quibbles and corrections that you can easily use to refine this basic prediction.) From this the probability of being a sum of two odd primes in 0 ways can be estimated, and turns out to go to 0 fast enough that we expect only a finite number of counter-examples. (2 and 4 are indeed counterexamples to the stronger statement, but we know of no others.)
This basic projection can be verified for small n by computer analysis, and indeed works out to be remarkably accurate. Therefore we have great confidence that Goldbach's conjecture is likely to be true, but absolutely no idea why.
The hope, some day, is to be able to close this gap. To not just make assertions based on known statistical facts about the distribution of primes, but to be able to prove them.
The best known example of a statement which should be true that we have not yet proven is the Riemann Hypothesis. It should be noted that Louis de Branges does claim to have proven it, nobody has investigated his claims depth, and he was right about the Bieberbach conjecture.
My favorite example of a statement which looks reasonable on the numerical evidence, cannot be supported by this kind of argument, which later proved false is Mertens conjecture.