Timeline for Arithmetic progressions of gaussian primes
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 23, 2020 at 8:23 | vote | accept | Augusto Santi | ||
| Aug 23, 2020 at 8:23 | vote | accept | Augusto Santi | ||
| Aug 23, 2020 at 8:23 | |||||
| Aug 22, 2020 at 17:39 | answer | added | Christian Elsholtz | timeline score: 2 | |
| Aug 21, 2020 at 4:59 | vote | accept | Augusto Santi | ||
| Aug 23, 2020 at 8:23 | |||||
| Aug 20, 2020 at 20:03 | answer | added | David Loeffler | timeline score: 6 | |
| Aug 20, 2020 at 20:03 | comment | added | David Loeffler | That wasn't the question that was asked. Moreover, you seem to be confusing "there is an arithmetic progression containing infinitely many primes" with "there are arbitrarily long (finite) arithmetic progressions consisting only of primes" | |
| Aug 20, 2020 at 19:57 | comment | added | Stanley Yao Xiao | @DavidLoeffler it doesn't, but inferring from the example given and the assumption that the OP understands how to count arithmetic progressions in the rational primes (which is what one gets if $u,v$ are co-prime rational integers and $n$ a rational integer parameter), I believe it is reasonable to assume that $u,v$ are not rational integers nor of the form $\pm ix$ for rational integral $x$. | |
| Aug 20, 2020 at 19:53 | comment | added | David Loeffler | Your assertion is still false -- what about 3i? Moreover, where in the question is it specified that $u, v$ can't be rational integers? | |
| Aug 20, 2020 at 19:51 | comment | added | Stanley Yao Xiao | Since a Gaussian integer $x = a + ib$ with $ab \ne 0$ is prime if and only if its norm is a rational prime, your question is equivalent to asking for fixed Gaussian integers $u,v$ whether exist infinitely many rational integers $n$ such that the expression $|u|^2 + 2n\Re(u\overline{v}) + n^2 |v|^2$, which is a quadratic polynomial in $n$ with rational integer coefficients, is prime infinitely often. There is no irreducible quadratic polynomial for which we know the answer. | |
| Aug 20, 2020 at 19:50 | comment | added | Stanley Yao Xiao | @DavidLoeffler I was working under the supposition that only primes that split in $\mathbb{Z}[i]$ count as "Gaussian primes", which I believe is intended by the question (typically the expression $u + nv$ given in the question will never equal a rational integer). I can edit the comment to clarify this | |
| Aug 20, 2020 at 19:48 | comment | added | David Loeffler | @StanleyYaoXiao Your claim about Gaussian primes is false (3 is a Gaussian prime, but its norm is 9). | |
| Aug 20, 2020 at 19:06 | history | asked | Augusto Santi | CC BY-SA 4.0 |