Timeline for Primitive roots
Current License: CC BY-SA 4.0
9 events
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| Nov 14, 2022 at 22:25 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Jan 28, 2011 at 17:54 | comment | added | Seva | I wonder whether there is a reasonably simple elementary argument showing that for a fixed, sufficiently large integer $N$, the set $[-N,N]\setminus\{0\}$ multiplicatively generates $({\mathbb Z}/p{\mathbb Z}$)^\times$ for ininitely many / positive proportion of all primes $p$. | |
| Jan 28, 2011 at 3:03 | comment | added | Joe Silverman | @Felipe: Amusing, I think we were composing our answers at the same time, with more-or-less the same material. | |
| Jan 28, 2011 at 2:45 | comment | added | Felipe Voloch | Heath-Brown's proof goes by first showing that given three primes, they will generate $(\mathbb{Z}/p)^*$ for infinitely many $p$ and current technology does not allow us to reduce the number of primes to two. His method was inspired by an analogous result of Gupta and Murty on elliptic curves, where you ask if a given set of $m$ rational points (satisfying some obvious necessary condition) generates the group mod $p$ for infinitely many $p$ and this can be proved as long as $m$ is big enough (5 for CM and 8 for non-CM? I forget). The conjecture would be $m=1$, but current tech is not enough. | |
| Jan 28, 2011 at 2:00 | comment | added | Zev Chonoles | However, $n_1,\ldots,n_k\in\mathbb{Z}$ could still generate $\mathbb{Z}/p\mathbb{Z}^\times$ even if none of them is a primitive root mod $p$. It seems like Hej's question might not require such hard results. | |
| Jan 28, 2011 at 1:58 | comment | added | Hej | Thank you. I was aware of that result. Maybe my question was not clear. So what I am asking is whether the group generated by 2 and 3 in $\mathbb{Z}_p^*$ is all of $\mathbb{Z}_p^*$. Equivalently, is it true that for infinitely many primes $p$, every number $m$ is congruent to some $2^a3^b$ mod $p$, with $a,b$ natural. | |
| Jan 28, 2011 at 1:56 | comment | added | Andrés E. Caicedo | There is also a nice presentation by Murty, "Artin's conjecture for primitive roots", the Mathematical Intelligencer, vol 10 (4) (1988), 59-67. | |
| Jan 28, 2011 at 1:55 | history | edited | GH from MO | CC BY-SA 2.5 |
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| Jan 28, 2011 at 1:50 | history | answered | GH from MO | CC BY-SA 2.5 |