Timeline for Quadratic Diophantine equations with all values prime
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 9, 2021 at 1:11 | comment | added | JoshuaZ | @NoamD.Elkies Doh on both parts. Thanks. | |
| Feb 8, 2021 at 23:56 | comment | added | Noam D. Elkies | For each k, I use Schinzel to find n, and thus D, and thus the equation. I can't apply Schinzel to a single Pell-type equation. (I could apply Schinzel to a parabola such as $y = 2x^2-1$.) And no, Green-Tao doesn't apply Dirichlet because Dirichlet says every arithmetic progression $a \bmod q$ contains infinitely many primes as long as $\gcd(a,q)=1$. | |
| Feb 8, 2021 at 23:52 | comment | added | JoshuaZ | @NoamD.Elkies I think I'm confused about something, possibly quantifiers here. Given such an equation isn't arbitrarily large collections imply infinitely many? (In the same sense that say Green Tao implies Dirichlet's theorem). | |
| Feb 8, 2021 at 15:11 | comment | added | Noam D. Elkies | Arbitrarily many, not infinitely many: the hypothesis would only give you arbitrarily large collections of prime values (with a different application for each $k$), not infinitely large. It's like arithmetic progressions of primes: a special case of Hypothesis H (now famously a theorem of Green and Tao) implies that there are arbitrarily long arithmetic progressions of primes, but there certainly aren't infinitely long ones! | |
| Feb 8, 2021 at 13:10 | comment | added | JoshuaZ | @NoamD.Elkies Hmm, if I'm following your reasoning then that maybe should suggest that the initial conjecture is false , since if Hypothesis H is correct then that should generate infinitely many prime pairs? Am I missing something? | |
| Feb 8, 2021 at 0:23 | comment | added | Noam D. Elkies | This probably follows from Schinzel's Hypothesis H, using a Pell equation $x^2 - D y^2 = 1$ with $D$ in one of the families such as $D = n^2 - 1$ for which the solutions $(x,y)$ are polynomials in $n$ (followed by some linear change of variable if needed to get irreducible polynomials with no predictable small factors). | |
| Feb 7, 2021 at 23:59 | comment | added | JoshuaZ | Hmm, this raises a related question: For any k can we find such an equation that has at least k prime solutions? | |
| Feb 7, 2021 at 22:39 | comment | added | Will Jagy | @Gerry, thanks. Just an example with concrete numbers. | |
| Feb 7, 2021 at 22:03 | comment | added | Gerry Myerson | @Wojowu the $w,v$ pairs are solutions of $w^2-2v^2=23$, and the factorizations of $w$ and $v$ indicate that there are a few solutions with both terms prime. | |
| Feb 7, 2021 at 21:58 | comment | added | Wojowu | I'm afraid I have no idea what this answer is trying to answer. | |
| Feb 7, 2021 at 21:54 | history | answered | Will Jagy | CC BY-SA 4.0 |