Menu
Last entry, I gave the expanded proof by cases of JF's second theorem, which implied that if two D's, D_x and D_y, shared a common factor, c, then at least one of the other prime factors, R_x and R_y, would be greater. For this entry, I will be skipping a section which, incidentally, JF says is very important. I plan on getting back to it later after I understand all of what JF's work does. This entry will correlate with the contents of the stream starting at about 1:29:37, which should be when the above video begins. This is where JF introduces what he calls the "T-set". Recall the equations we have been examining previously. The equations of the form: And, again, we are supposing the set of D's produced consist of prime factors which can be put back into the equation above, the reiterative search for a pair of primes. Now, we can slightly modify this equation wherein we can likewise introduce these corresponding R's into. This is what produces the T-set. Here is that equation: Now, we know by JFT for the D equation that D and E will always be coprime. As you can probably guess, it is the exact same demonstration which shows that T will likewise always be coprime to E: Like the proof before, R must be prime, which implies no R/C exists which is an integer, and since B must be an integer, we can conclude the contrary is true. Therefore, T is always coprime to E. Now it has been established that both T and D are coprime with E. The next natural question you would have to ask is whether or not T and D are coprime with each other. In the stream, JF says that they are coprime except for if they have a common factor of 3, or in the "+" case. JF has a different proof, but I will demonstrate to the reader now some of these truly interesting results which were on the stream in more detail: This will lead you to two expressions for R: Expression 1: Expression 2: Now, here is where things get interesting. We can do the same type of demonstration on expression 2 as we have been doing many times before. That is, assume that T and D have some common factor C. By letting T= AC and D = BC: Another spicy contradiction. But what does this imply? In expression 2, since R is positive, we can conclude that T > 2D. But this may not always be the true. A case in point: Suppose E = 224 and R = 17. We will use the addition side of the D equation:
So, were you to input these numbers, all of which satisfy the constraints, into expression 2, you would end up with a negative number. This means that the two results should be treated as cases. Expression 1 is the case for when 2D > T and expression 2, conversely, when T > 2D. This leads us to make some very curious implications: So, let us suppose you have case 1, where 2D < T: you know T and D necessarily share no prime factors. For case 2, we can actually use my above case in point: 129 (mod 3) = 0, so T, which is 207, will likewise share the common factor of 3. (For the record: 207 = 3 * 69). For case 3, I can offer you another case in point:
And neither 149 nor 187 are divisible by 3. But 2D-T will always be divisible by 3. Next, I will show you a series of three graphs: We can draw some important facts looking at these three graphs. All are of the same forms as the respective either D equations or the single T equation. Facts we have found:
That is, the only time you will ever find a common factor between a D_x and a T_x is when D_x is found from: E/2 + R_x = D_x. And R will only be in a domain of (0, E/4). Not only will those two facts be true, but the only common factor they will ever have will be 3. Wow. That does not imply all primes in this domain yield D's and T's divisible by 3, but if they exist, they can only be found there. A similarly important fact which is not implied here is that a D_x cannot share any common factors with a T_y. I say again, that is not necessarily true, they could potentially share common factors in that case. Maybe D_2 shares a common factor with T_4. That is a post for another time, though.
I was proud of this post. Seems like some good information could be within it. Until later, - AF
0 Comments
Leave a Reply. |
AuthorI'm just trying to learn about everything I can. Archives
June 2020
Categories
All
|