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I wanted to expand on some of the mathematics covered on one of JF's livestreams. That way, there is at least a more, uh, clean presentation of what JF wrote down. I am just clarifying it in my own way, maybe making some interjections of my own, and so on. The video is here: So, the Goldbach Conjecture: "Every even integer greater than 2 can be expressed as the sum of two primes." -From Wikipedia And I will list the constraints to JF's theorem: Now, at around 1:14:07 JF's writing gets messy, but he says the numbers produced by the following equation: are prime pairs which add to E. But, as he later elaborates on, this is not always true. A case in point is let E = 28, and let R_1 = 5. This produces 9 and 19. If you used bigger numbers, then maybe you could get a pair from R_1 and one of the D's produced, here you cannot. You can see here, 9 is not a prime number. But he does this reiterative approach. So we know that 9 is coprime, and obviously consists of a pair of prime factors. In fact, if we take that pair of prime factors, 3 and 3, we can let R_2 = 3 here: And JF's algorithm nails it this second time: since The D_2's generated are 17 and 11! So we reach a conclusion for the first iteration, that is the pair of D_1's: Either the D pairs produced are prime and sum to E, or they consist of prime factors, which could be put back into the search, using JF's algorithm. I cannot go so far as to say that the prime factors produced will lead to a pair of primes which sum to E, since that would be a proof of the Goldbach conjecture, but I can say, at least, that the prime factors can go in, since, by definition, they would be prime, and they would be less than R_1, and consequently do-able as R_2. All of that is well and good, but just what is JF's Theorem, or maybe, if you prefer, Gariepy's Theorem? It is for the general equation: Given the aforementioned constraints, (E divisible by 4, R is a prime where E is coprime to R), JF's theorem says that D is also coprime with E. There exists a proof, which can be demonstrated by contradiction. JF has his own basically identical approach, but I can give you a similar idea here:
Now we will multiply both sides by the reciprocal of c: But by definition, R is prime, a and b are also integers, so there is a contradiction. Therefore, we can say that JF's Theorem, that is, for the general equation, D is always coprime with E, is true. Congratulations, JF! You have your own theorem. That is actually pretty cool.
We also know, incidentally, that D will be an odd number, since E is divisible by 4, and there can be no sum of two prime numbers in the range we are looking for (i.e. E < 4) where the sum can be expressed by 2 + x. That is the proof of JF's Theorem. I know that is not all of his work he presented, but this entry was long, so I want to add a second entry later. Just to make it more discrete of a resource, and not a mountain of information. Sorry about the LaTeX being shown in gifs, also. Weebly is pretty messy at formatting, and images were to only clean way I found for putting the equations on here. - AF
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