Topics Mathematics

Question Goldbach's conjecture states that every even integer greater than two can be expressed as a sum of two primes. There is no formal proof of this conjecture. However, every even integer greater than two has been shown to be a sum of two primes once we started looking. Is this acceptable justification for believing Goldbach's conjecture? Can we determine mathematical theorems based on observational evidence?

Accepted:
April 20, 2011

Comments

Acceptable to whom? I don't think the evidence you provide would or should convince mathematicians. They justify their beliefs about conjectures like this by appeal to proofs or counter-examples. So long as neither is forthcoming, they will rightly suspend belief.

But for the rest of us, perhaps the kind of "observational" evidence you suggest might be convincing. Here it would not help much to argue that, because we have found Goldbach's conjecture to be correct up to 10^n, it is probably correct all the way up. One reason this would be unhelpful is that the as yet unexamined even numbers are infinitely more numerous than the examined ones, so we will always have examined only an infinitesimally small sample. Another reason this would be unhelpful is that the examined even numbers are not a representative sample -- rather, they are all very much on the small side, as far as numbers go.

So the probabilistic argument would have to go differently. Let's first establish that the two prime numbers we're looking for are both odd. There is only one case where this is not true -- 2+2=4 -- but we can just put this case aside as settled.

Now examining any even number involves writing it, in as many ways as is possible, as the sum of two odd numbers and then looking through all these sums to find one where both of the odd numbers are primes. Obviously, the larger even numbers get, the more different options there are of writing this number as the sum of two odd numbers. When the number in question is 2n, then we have about n/2-1 such options. I say "about", because the number of options is the same for any number divisible by 4 and the even number just below it. Thus we have 2 distinct options of writing 12 as the sum of two odd numbers (3+9, 5+7) and 2 distinct options of writing 10 as the sum of two odd numbers (3+7, 5+5). For any large even number, the number of decomposition options is about one quarter of this number. So it looks like, as we get to larger and larger even numbers, the probability that each can be written as the sum of two primes gets larger and larger, because there are more and more options. This probabilistic reasoning assumes that the probability that any given way of writing some even number as the sum of two odd numbers (E=O1+O2) is such that O1 and O2 are both primes can be estimated as the square of the probability that any odd number smaller than E is prime.

To illustrate, let the even number be 100. There are 50 odd numbers below 100 and of these 24 are prime. So the probability that any randomly selected odd number below 100 is prime is 0.48. And the probability that any two randomly selected odd numbers below 100 are both prime is then 0.48^2 or about 0.23. So the probability that any two randomly selected odd numbers below 100 fail to be both prime is about 0.77. But this substantial probability now gets whittled down by the fact that there are many options for writing 100 as the sum of two odd numbers -- in fact, there are 25 such options. And we may then surmise that the probability that none of these options involves a pair of primes is about 0.77^25, or about one in 700, or about one seventh of 1%, or about 0.0014.

Let's repeat the exercise for E=1000. There are 500 odd numbers below 1000 and of these 167 are prime. So the probability that any randomly selected odd number below 1000 is prime is about one third. And the probability that any two randomly selected odd numbers below 1000 are both prime is then about one ninth. So the probability that any two randomly selected odd numbers below 1000 fail to be both prime is about eight ninth or 0.89. But there are 250 options for writing 1000 as the sum of two odd numbers. And the surmised probability that none of these options involves a pair of primes is then about 0.89^250. This amounts to one chance in 6 trillion, or a probability of about 0.00000000000016.

We see here that, as we go from E=100 to E=1000, the probability of finding no decomposition into two primes falls off very steeply: going from 100 to 1000, the probability was cut by a factor of about 10 billion. Just to be sure, let's repeat the exercise one more time for E=10000. There are 5000 odd numbers below 10000 and of these 1228 are prime. So the probability that any randomly selected odd number below 10000 is prime is about one quarter. And the probability that any two randomly selected odd numbers below 1000 are both prime is then about 6% or 0.06. So the probability that any two randomly selected odd numbers below 1000 fail to be both prime is about 0.94. But there are 2500 options for writing 10000 as the sum of two odd numbers. And the surmised probability that none of these options involves a pair of primes is then about 0.94^2500. This amounts to one chance in 15 times 10^66, or to a probability of about 0.000000000000000000000000000000000000000000000000000000000000000000066.

Note that, as we went from E=1000 to E=10000, the probability of finding no decomposition into two primes has fallen off even much more steeply (than in the earlier move from E=100 to E=1000): going from 1000 to 10000, the probability was cut by a factor of about 2.5 times 10^54!

Now we must also consider one other factor here. Let's think of E=100, E=1000, E=10000, and so on as centers of "neighborhoods". So defined, these neighborhoods become larger as we go up. There are about 140 even numbers in the E=100 neighborhood (those between 31 and 310, roughly) and about 1400 even numbers in the E=1000 neighborhood (those between 310 and 3100, roughly), and so on. How does this factor affect the calculation? Again, estimating very roughly, if the chance that any given even number around 100 is decomposable into two primes is about 0.9986, then the chance that all 140 of them are so decomposable is about 82% or 0.82. If the chance that any given even number around 1o00 is decomposable into two primes is about 0.99999999999984, then the chance that all 1400 of them are so decomposable is about 0.99999999999984^1400, or about 99.999999977%. And if the chance that any given even number around 10000 is decomposable into two primes is about 0.999999999999999999999999999999999999999999999999999999999999999999934, then the chance that all 14000 of them are so decomposable is about 0.999999999999999999999999999999999999999999999999999999999999999999934^14000, or about 0.999999999999999999999999999999999999999999999999999999999999999.

As these very rough calculations show, the probability of finding a counterexample declines very (and increasingly) steeply as we get to higher neighborhoods. To be sure, there is a non-zero probability in each neighborhood, and there are infinitely many neighborhoods. But this still does not add up to a substantial probability -- much like the infinite series of $1 + $0.1 + $0.01 + $0.001 + ... does not add up to a fortune.

I conclude then that, if there's no counter-example to Goldbach's conjecture in the lower neighborhoods, then this conjecture is extremely likely to be correct. My support for this conclusion rest on a kind of (quick and dirty) probabilistic reasoning that perhaps fits what you have in mind when you speak of "observational evidence" (I "observed" how many prime numbers there are in this and that neighborhood, and so on). I find this reasoning compelling, so am starting herewith to believe that Goldberg's conjecture is true. But I would not be surprised at all if mathematicians were entirely unmoved by this sort of reasoning.