Just studying the paper PRIMES is in P
, although I've tried great efforts, some proofs are still not so clear(or obvious) to me, especially the proof of Lemma 4.3
. The problem is by choosing r
the smallest number that does not divide the product $n^{\lfloor logB \rfloor}\prod_{i=1}^{\lfloor log^2n\rfloor}{(n^i-1)}$, why the fact $(r, n)$ cannot be divisible by all the prime divisors of r
holds? And why if violated, r
will divide $n^{\lfloor logB \rfloor}$? Any relation with the observation "the largest value of $k$ for any number of the form $m^k \leq B$ , for $m > 2$ is $\lfloor logB \rfloor$"? And finally, the conclude of this proof, I cannot find trivial relation between $r$ and $B$, any theorem or fact I miss?
Thanks and Best Regards!
EDIT:
Follow Will's advice below, I found the article linked above is a draft and referred to the published version. Much more clear indeed! However, there is still one point I'm not sure:
At nearly the end of proof to lemma 4.3, the authors say "If $(s,n)>1$, then since $s$ does not divide $n$ and $(s,n)\in {r_1,r_2,…,r_t }$" then "$r=\frac{s}{(s,n)}\not\in {r_1,r_2,…,r_t }$". Where $r_i$ is defined as numbers either $o_r(n)\leq \log^{2} n$(which is the bound we want to approach, I think) or $r_i$ divides $n$ and $s$ is a number $\leq \lceil \log^{5}n\rceil$ such that $s\neq r_i$(previous lemma shows we can always find such an $s$ under the hypothesis). Why there is no pobability $o_r(n)\leq \log^{2}n$? In which case, seems no assumptions are voilated.
Thank you!