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In post 7, we went through the proof by cases, which was a contradiction of R_x or R_y being greater than a common factor, c, in the stream at 1:39:50. JF does suppose that c will be a prime, and I had also reasoned that in post 7. But, after discussing it with him, we realized that it could just be a trivial assumption, and not necessarily true. So how can we know that c is actually prime? It could be because I have not slept in two days, or that the approach was not intuitive for me, but I, after quite some time, believe we can demonstrate the c is necessarily prime. I also wanted to elaborate on a conclusion I had reached before in post 7, which I will do first. This special case of z = 1 is applicable in the two cases: And, from the work we did in post 7, we know: The problem was one of the conclusions I reached, which was that if z = 1, then 2D_x > T_y. That is true, but I felt the need to mention that this does not necessarily imply that 2R_x > R_y. That is because there were two cases where z could be 1. That is all, but it could lead to some possible misconceptions. With that out of the way, what is a method we can use to determine that c is a prime? We can employ the proof by contradiction again. Suppose that c is comprised of prime factors such that c = ab. So 2D_x will be divisible by a, b and c. Perhaps you could make the case that if: But then you would need to recall: This leads to a problem already. First, neither a nor b would not be applicable, since z must be 1 for the case to work. But, then the factor, a, would be a common factor for 2D_x - T_y and also R_y +/- 2R_x. You can apply this reasoning and you will arrive at the contradiction: So I think we are safe to say it, JF. We have z-primacyI will add more content later. Keep your eyes out, lads.
- AF
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