Timeline for Factorizations of cyclotomic polynomials valuated at primes
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11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 24, 2023 at 3:15 | vote | accept | Pablo Spiga | ||
| Sep 23, 2023 at 19:14 | comment | added | so-called friend Don | @GHfromMO Done. | |
| Sep 23, 2023 at 19:13 | answer | added | so-called friend Don | timeline score: 10 | |
| Sep 23, 2023 at 18:25 | comment | added | GH from MO | @so-calledfriendDon I suggest that you turn your comment into an answer so that this question can be closed. | |
| Sep 23, 2023 at 3:39 | comment | added | Pablo Spiga | Great!!! This is what i needed! Thanks. | |
| Sep 22, 2023 at 21:45 | comment | added | so-called friend Don | Following up on GH from MO's comment, what you want follows from work of Siegel; a quantiative version could be deduced from this paper of Shorey and Tijdeman: numdam.org/item/CM_1976__33_2_187_0 | |
| Sep 22, 2023 at 18:17 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 22, 2023 at 18:16 | comment | added | GH from MO | What you really want to prove is this: for fixed $n$, the largest prime factor of $\Phi_n(p)$ tends to infinity (as $p$ runs through the primes). | |
| Sep 22, 2023 at 17:40 | comment | added | Pablo Spiga | I did not know the definition of smooth number, I've just looked that up. You are right that there is a connection to smoothness. Using this terminology I am asking to prove that, for $n \ge 3$, either $\Phi_n(p)$ is not $c$-smooth, or $p$ is small (in terms of $c$). I do not see how the results if Stewart and Yu are helpful here. | |
| Sep 22, 2023 at 16:51 | comment | added | Stanley Yao Xiao♦ | If I am interpreting the question correctly, what you are asking is that either $\Phi_n(p)$ is not $g(p)$-smooth for some function $g$ tending to infinity, and to give some sort of growth rate for $g$. I believe this is possible using some type of quantitative analogue of the $abc$-conjecture, for example due to Stewart and Yu. | |
| Sep 22, 2023 at 15:57 | history | asked | Pablo Spiga | CC BY-SA 4.0 |