Timeline for Relation between elements with fixed exponent over different $\mathbb{Z}^\times_p$
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 9 at 14:27 | comment | added | Somudro Gupto | @MaxAlekseyev Thanks for this calrification. It is correct. | |
| Apr 9 at 11:17 | comment | added | Max Alekseyev | For $k=(p_i-1)/2$ (and its odd multiples), we have $x_i=p_i-1$, which shows that no better upper bound is possible. | |
| Apr 9 at 10:39 | comment | added | Somudro Gupto | @MaxAlekseyev Yes, $2<x_i< p_i$ trivially. An $\mathcal{O}(\log{p_i})$ bound would be more meaningful, | |
| Apr 9 at 1:52 | comment | added | Max Alekseyev | Isn't $p_i$ a readily available upper bound for $x_i$? | |
| Apr 8 at 19:56 | comment | added | Somudro Gupto | Yes, that is correct. Thank you again. It could be really helpful if this fraction could be expressed as a function of $N$. I think it needs a separate question. | |
| Apr 8 at 19:22 | comment | added | Will Sawin | Not necessarily a non-negligible fraction in the sense of positive density, but I would expect this does happen for infinitely many primes. | |
| Apr 8 at 18:47 | comment | added | Somudro Gupto | Thank you professor @WillSawin for this answer. If I understood correctly, in other words do you want say that $x_i=3$ for a non-negligible fraction of primes that have $2$ as one of their primitive roots? | |
| Apr 8 at 17:28 | comment | added | Will Sawin | If I understand your question right, $x_i \geq 3$ should be the best possible lower bound (since $1$ and $2$ are impossible or obvious reasons). We can find examples where this bound is sharp by factoring $2^k-3$ - many prime factors should have $2$ as a primitive root. Some examples with $k/\log p$ small are $5,13,29, 61, 509$. | |
| Apr 8 at 17:12 | history | edited | Somudro Gupto | CC BY-SA 4.0 |
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| Apr 8 at 16:43 | history | edited | Somudro Gupto | CC BY-SA 4.0 |
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| Apr 8 at 16:43 | history | edited | Somudro Gupto | CC BY-SA 4.0 |
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| Apr 8 at 16:42 | history | edited | Somudro Gupto | CC BY-SA 4.0 |
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| Apr 8 at 16:40 | history | edited | Somudro Gupto | CC BY-SA 4.0 |
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| S Apr 8 at 16:40 | review | First questions | |||
| Apr 8 at 19:56 | |||||
| S Apr 8 at 16:40 | history | asked | Somudro Gupto | CC BY-SA 4.0 |