Timeline for Quadratic non-residues in elliptic divisibility sequences
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Aug 7, 2020 at 20:12 | comment | added | Dror Speiser | Still, heuristically the number of such $p$ should be finite: each $p$ divides $d(n_pP)$ for some $n_p$, and then there's periodicity. $\left(\frac{d(nP)}{p}\right)=1$ for all $n<n_p$ occurs with chance $2^{n_p}$. So we can estimate an upper bound for the number of "bad" primes: $\sum_n \omega(d(nP))/2^n$, where $\omega(k)$ is the number of distinct primes dividing $k$. Since $d(nP)=O(e^{cn^2})$ for some $c$, and $\omega(k)=O(\log{k})$, this sum is finite. | |
| S Aug 7, 2020 at 10:00 | history | bounty ended | CommunityBot | ||
| S Aug 7, 2020 at 10:00 | history | notice removed | CommunityBot | ||
| Aug 5, 2020 at 2:06 | comment | added | Jeremy Rouse | It appears that for a fixed $E$ and $P$, the primes for which there is no $n$ so that $\left(\frac{d(nP)}{p}\right) = -1$ can be reasonably large. I just looked at the example of $E : y^{2} = x^{3} - 2x$ and $P = (2,2)$, and found that there is no such $n$ for $p = 17$, $257$, $1009$, $1361$, $26881$, and $141041$. | |
| Aug 4, 2020 at 8:56 | comment | added | Daniel Loughran | We already have some ideas how to prove this "density zero" result, so are really looking for a proof that there are only finitely many exceptions. | |
| Aug 3, 2020 at 8:38 | history | edited | Daniel Loughran | CC BY-SA 4.0 |
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| Jul 30, 2020 at 17:08 | comment | added | Dror Speiser | @Stanley Yao Xiao up to some work reconciling the definition difference, I think so: combine the main result in Reductions of Points on Elliptic Curves by Akbary et al, with the main result in Character Sums with Division Polynomials by Shparlinski and Stange. | |
| Jul 30, 2020 at 12:03 | comment | added | Stanley Yao Xiao | Is it even known that the primes excluded have density zero? | |
| S Jul 30, 2020 at 8:44 | history | bounty started | Daniel Loughran | ||
| S Jul 30, 2020 at 8:44 | history | notice added | Daniel Loughran | Draw attention | |
| Jul 29, 2020 at 6:41 | history | edited | Daniel Loughran | CC BY-SA 4.0 |
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| Jul 28, 2020 at 8:22 | comment | added | Daniel Loughran | Good question. I don't know. | |
| Jul 27, 2020 at 20:15 | comment | added | Dror Speiser | Is this known for Lucas sequences? | |
| Jul 27, 2020 at 16:15 | history | asked | Daniel Loughran | CC BY-SA 4.0 |