Timeline for Divisibility in products of prime numbers
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 3, 2012 at 6:29 | answer | added | Aaron Meyerowitz | timeline score: 1 | |
| Feb 1, 2012 at 16:40 | comment | added | Valerio Capraro | so, the conjecture would be that the set of $k$'s such that $h_k$ solves that congruence has density $\frac{1}{(p_1-1)\cdots(p_i-1)}$? This would be nice! | |
| Feb 1, 2012 at 16:10 | comment | added | user9072 | ...a finite abelian group with a 'new' and 'random' element. Eventually this 'new' element should just happen to be the inverse of the product of the preceeding ones (at each step one has a constant probabilty for this to happen). I have however no idea if one can make this rigorous. Yet what is definitely the case is that after some explict number of steps one has a collison of an hk with an hk' (as there are only fin many choices) and 'cancelling' the common start one gets what I claimed in an elementary way. | |
| Feb 1, 2012 at 16:06 | comment | added | user9072 | Perhaps I should add something to my preceeding comment as otherwise it could seem silly (taking k=0 if one allows and just applying Dirichlet; or some modification of this argument). If one defines rj as the residue of p(i+j) modulo the product of the first i primes, then all the rj are invertible classes. So one has an infinite sequecence in the finite group of invertible residue classes modulo the product. Except some bias due to size at the start these elements should be essentially 'random'. So the hj arises by multiplying some element of... | |
| Feb 1, 2012 at 15:53 | comment | added | user9072 | Not sure if this can be proved but I would be surprised if it were not true. Perhaps you know this already, but for h(j,k) the product of the i + j th to the i +j + k th prime there would be a short argument that show h(j,k) will be congruent one (even with bounds on j and k). | |
| Feb 1, 2012 at 15:22 | history | asked | dama | CC BY-SA 3.0 |