Timeline for Primes of the form $d^2+d+1$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 19, 2023 at 11:03 | vote | accept | numberwat | ||
| Jan 18, 2023 at 17:13 | comment | added | Sylvain JULIEN | I guess so, that's the goal actually. An approach through dynamic systems might be fruitful. I have the vague idea that this could be related to hypothetical "n-th roots of automorphisms" i.e. maps $g$ such that $g^{\circ n}$ is an automorphism. | |
| Jan 18, 2023 at 17:05 | comment | added | Steven Stadnicki | @SylvainJULIEN Wouldn't that immediately imply this specific case of Bunyanakovsky? | |
| Jan 18, 2023 at 17:02 | comment | added | Sylvain JULIEN | It might be interesting to restrict oneself to the case where $d$ is itself prime and study the images of the iterates of $f:x\mapsto x^2+x+1$, for example trying to establish that for infinitely many primes $p$, there exists an integer $k_{p}$ such that $f^{\circ k_{p}}(p)$ is prime. | |
| Jan 18, 2023 at 15:45 | answer | added | Stanley Yao Xiao | timeline score: 7 | |
| Jan 18, 2023 at 15:12 | history | edited | Steven Landsburg | CC BY-SA 4.0 |
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| Jan 18, 2023 at 14:32 | comment | added | Fedor Petrov | I am afraid that it is not even known whether a polynomial of degree more than 1 for which it holds exists | |
| Jan 18, 2023 at 14:13 | comment | added | Wlod AA | In the given context it's good to recall en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem. | |
| Jan 18, 2023 at 13:45 | comment | added | Boaz Moerman | @JoséHdz.Stgo. This is true for every polynomial though, so it is not a special property of this one. This is for example shown in the answer to math.stackexchange.com/questions/86018/…. | |
| Jan 18, 2023 at 13:40 | comment | added | José Hdz. Stgo. | At least you can prove that if $X$ is an integer greater that $1$ such that $p(X)$ is a prime number then $X \equiv 0 \pmod{3}$ or $X \equiv 1 \pmod{3}$: in other words, the polynomial does produce infinitely many composite numbers. | |
| Jan 18, 2023 at 13:40 | comment | added | Boaz Moerman | As mentioned in the section "Partial results: only Dirichlet's theorem" of the Wikipedia page you linked, this is only known for linear polynomials, and for no others. | |
| Jan 18, 2023 at 13:33 | comment | added | YCor | I think there is no polynomial of degree $\ge 2$ for which this is known to hold. | |
| Jan 18, 2023 at 13:28 | history | asked | numberwat | CC BY-SA 4.0 |