Some Thoughts About Prime Numbers

Why Goldbach is probably unprovable and Twin Primes is probably true

https://www.researchgate.net/figure/This-spiral-represents-all-ordinal-numbers-less-than-o-o-3_fig1_344155343

Once upon a time I studied some math (mostly physics math, i.e. linear algebra, differential geometry, group theory, etc.). In the limit of free time going to infinity, one will eventually find oneself contemplating unsolved number theory problems like the twin primes conjecture. Not being a number theorist, nor possessing so much as an antiquated Dover book on number theory, I don’t expect such contemplation to be of much interest to professional integerometers, but perhaps an aspiring practitioner of impractical mathematics may find it insightful.

Twin Prime Conjecture Twin primes are pairs of prime numbers with a difference of two, such as 17 and 19. The twin prime conjecture claims that the number of twin primes is infinite. My naive suspcion is that this conjecture is true. Consider the proof that the set of primes is infinite. If there were a finite number of primes then we could take the product of every prime and add (or subtract) one. None of the factors which we used to construct this number will be a factor (if we divide ab+1 by a then we get b+1/a which is not an integer), and so it must be a prime bigger than the biggest prime, hence there cannot be a biggest prime. QED. Using this same construction we can find candidates for twin primes, e.g. 2(3) gives us {5,7} and 2(3)(5) gives us {29,31}. The twin primes conjecture would be a straightforward corrolary if we could prove that the gap between primes outpaced the growth of prime products (Goldbach’s conjecture would also be trivially false if such enormous prime gaps existed). Otherwise, we inevitably end up with numbers like 2(3)(5)(7)-1 = 209 which has 11 and 13 as factors.

It is well established in analytic number theory that the number of primes less than x grows like x/ln x (establishing this is what Riemann was doing when he came up with his infamous conjecture). Approximate growth, though, will not be sufficient for an actual deductive proof or counterproof of the twin primes conjecture. What we can do instead is find a lower bound for the number of primes between, say, p and 2p where p is some prime number (an interval particularly relevant to the Goldbach conjecture since one of the primes in the sum must be between half the number of interest and the number itself). To do this we first treat every number as a potential prime. Next we eliminate half of the numbers, i.e. the numbers which have 2 as a factor. Now we remove one third of the remaining numbers. Then one fifth, and so on up to the prime immediately preceding p. What we end up with is a familiar looking product: p[prod(1-1/q)] (I apologize that I am not able to use LATEX and must write equations in plaintext). It is a simple matter (left as an exercise for the reader) to write this product as a recursion relation and show that it is strictly increasing (though slower than the interval size). To see that this is a lower bound we can use, say, p=251 (the 54th prime) to calculate 25 primes between 251 and 501 which is significantly less than the 42 actual primes. What happened is that we overcounted when we eliminated bigger composite numbers, but overcounting is fine since what we wanted was a lower bound rather than an accurate estimate.

There are a few ways that a more competent number theorist than myself might go from here. The large number of primes makes collisions with the Euclid construction inevitable, but we might try to show that an infinite sequence of collisions cannot happen (and thus that we will, eventually, arrive at another pair of twin primes). Or maybe we could show that because the primes cannot be evenly distributed (if they were accidently evenly distributed in one bracket then they would necessarily not be evenly distributed in the next overlapping bracket) and the maximum gap between primes grows fairly slowly that the oscillating gaps will always, like a sort of quasi-bounded random walk, be two from time to time. Proving is, of course, easier said than done.

So, why do I believe that the twin prime conjecture is true? Let’s go back to the Euclid construction and pretend that the set of primes is finite. Now we can imagine a wheel for each prime number with 2(3)(5)... spokes and 1/p of the spokes covered. Overlay the wheels and the uncovered spokes will be relative primes. Such a wheel will always have twin primes which will always repeat indefinitely as we traverse the wheel again and again. Even as we add new wheels, the gaps between spokes become larger and the overlap with existing holes will be effectively random (prime numbers are almost by definition randomly distributed, since patterns follow from repetition); if 11 happens to accidently cover 209 then we merely have to spin the wheel again and we know that we will be safe from 11 for ten spins (then we get 2^11(3)(5)(7)-1 where 11 is again a factor). The randomness of gaps and slowness of their increase seems to suggest that no matter how high we go we will eventually get another pair of twin primes.

Goldbach’s Conjection Goldbach’s conjecture claims that every even integer can be written as a sum of two primes. My semi-informed opinion is that Goldbach is true "almost everywhere" (to the extent that this expression is meaningful for integers) but that there is no non-probabilistic reason why it should be universally true.

Start by assuming that primes are distributed in a random fashion with a 1/log x density (which we could weaken by assuming instead an unspecified slower-than-exponential decrease, per the lower bound that we found above). We can visualize Goldbach’s conjecture as a cribbage board whose holes represent odd integers which increase to m/2 at the finish and then follow the adjacent track to reach m at the start, where m is any even positive integer (and m/2 is double counted if it is odd). If we place a peg in every prime number hole then Goldbach is equivalent to every even number having adjacent pegs somewhere on the board. By assuming a random distribution we have turned this into a combinotronics problem: what fraction of combinations (or "microstates") has no adjacent pegs and what is the probability that a number will eventually be such a combination?

Intuition could lead us astray in this problem. Imagine that we have two holes and one peg on each side. The probability of a collision is 1/2. Now double the size of the board while maintaining the 1:2 peg:hole ratio (i.e. assume that pegs increase linearly). The probability of a collision is now 5/6. What this shows is that the number of pegs affects collision probability more than the number of holes. How much more? There are nCp (n choose p) ways to arrange the pegs on the right track and (n-q)Cp collision-free combinations, where n = m/4 and q,p are the number of pegs in the left,right tracks. Dividing the former by the latter gives us (n-q)!(n-p)!/n!(n-q-p)!. It is not enough for q/n and p/n to go to zero for this ratio to go to one because q and p must decrease quickly enough to negate the factorials or else the n! in the denominator wil rapidly dominate everything else. What we need is for p (and thus q as well) to eventually stop growing and approach a constant P. This will give us a Pth order polynomial in both the numerator and denominator which will indeed go to 1 as n goes to infinity. So arctan(n) would work, as would 1-exp(-n). What is important, though, is that n/ln n definitely will NOT work. The density of primes decreases with n, but not enough to overcome all the new combinations that we can make with every new prime. So as our numbers become bigger, the probability that a particular number will violate Goldbach’s conjecture diminishes.

Maybe the probability of any one number violating Goldbach is infinitesimal but the probability that some number will eventually violate Goldbach is finite so that we might, with a finite amount of resources, eventually find a counterexample. We can check this by multiplying all the probabilities of collision (which is 1 minus the probability of no collision) as an infinite product. We don’t need to actually calculate this product, we just want to know if it converges to 1 or 0 or somewhere in between. If the first few terms are significantly less than 1 then the product obviously will not converge to 1 even if it doesn’t converge to 0. What we can do, though, is manually check some arbitrary finite number of terms and then set these probabilities equal to 1 and thus, provided the product does not converge to 0, we can get arbitrarily close to 1. We can expand the infinite product into 1 minus an infinite sum of no-collision probabilities plus additional terms which alternate in sign and will be smaller than the infinite sum if it converges, which it does (use your favorite comparison test from calculus II). As we check numbers and remove them from the infinite sum it’s value gets smaller and smaller and the probability of every number having a collision goes to 1. In other words, if there is a needle in our infinite haystack and it is not in a sufficiently large finite set of numbers which we have manually checked then it is extremely unlikely to be in the remaining infinite set of numbers.

What we will do now, which maybe we should have done first, is "prove" that Goldbach is unprovable if true. We do this because if it were provable then it probably would have been proven by now. How we do this is by first assuming that Goldbach is true and provable and that we have proven it to be true without specifying how we did this. Next we assume that we have some number m whose cribbage board is one step removed from having no collisions (call this G(1)), i.e. that m+2 will violate Goldbach if m+1 is not a prime number. We know, then, that m+1 must be prime because we know, by assumption, that Goldbach is true. This, however, creates a contradiction. In order to establish that m+1 is prime we need to know every prime number up to sqrt(m+1) (actually, we don’t, but this is more of a first draft of a proof than an actual factual proof). But to determine that a number is G(1) we only need to know the primality of numbers on the right side of the cribbage board, i.e. the primes between (m+1)/2 and m+1. We also need to know that adjacent numbers on the left will be composite, but this is moderately simpler than determing that a number is not composite, i.e. prime. Because the existence of a G(1) number would allow us to predict primality with less information than we actually need to determine if a number is prime we can conclude that G(1) numbers cannot exist.

Now we will prove by induction that a G(k) number cannot exist for any number k, i.e. that even numbers do not exist. Assume that G(k) numbers are not allowed. For the same reason as above, a G(k+1) board would allow us to predict that the next number must be prime lest we arrive at an impermissible G(k) board. Thus, our assumption that we were able to prove Goldbach to be true leads us to the conclusion that a number cannot be any number of steps removed from a Goldbach-violating number, which means our assumption was false, because even numbers obviously exist. What this means is that the Goldbach conjecture is either false or true but unprovable. And what this means is that if Goldbach is false then the only way to prove that would be to find a contradiction (we can’t assume that it is true and prove by contradiction because assuming that an unprovably true statement is true gives us no information). But we showed above that the odds of finding such a contradiction are practically zero. So Goldbach, if the above reasoning is sound, is effectively a lost cause.

Conclusion Number theory problems are interesting because they are easy to state but difficult to prove, though a proof often seems to be almost graspable but just out of reach. Attempting to prove something, even if unsuccessful, can be enlightening; even if the puzzle remains unsolved, maybe after your attempt you will have discovered something new and interesting about numbers. The above proof that Goldbach’s conjecture is unprovable if true, for instance, is not correct because it relies on a false assumption that determining that a number p is prime requires knowing all the primes up to sqrt(p). Nevertheless, I still decided to include it because it is interesting to think about and I still like the direction it tried to go in; it still seems to me that there is no justifiable reason why a board which is accidentally one peg away from violating Goldbach should determine whether the next number will be prime.

The twin prime conjecture may still have some hope of being solved one day, but my opinion about Goldbach’s conjecture is that it is not really an interesting problem. Postulate that a royal flush cannot be obtained in draw poker and you will never obtain a counterexample by brute force. Such a situation follows not from some interesting underlying property of numbers but merely from the extremely large number of ways to not make a royal flush in draw poker compared to the very few ways to make one. Goldbach has the additional problem that the set of combinations actually increases as more numbers are tested. Still, Goldbach seems to me to follow more from the largeness of factorials in combinotronics problems than from an interesting underlying feature of prime numbers. I shall continue to think more about it while I still have abundant free time to do so.