Timeline for Extension of Dirichlet's Arithmetic Progression Theorem
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1, 2018 at 16:34 | vote | accept | JMP | ||
| Dec 1, 2018 at 13:48 | comment | added | KConrad | The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers". | |
| Dec 1, 2018 at 13:44 | comment | added | KConrad | You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), \ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)\cdots f_r(x)$ is not identically 0 on $\mathbf Z/(p)$. | |
| Dec 1, 2018 at 10:16 | answer | added | Taras Banakh | timeline score: 7 | |
| Dec 1, 2018 at 9:54 | history | asked | JMP | CC BY-SA 4.0 |