Timeline for Set of quadratic forms that represents all primes
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2020 at 12:58 | vote | accept | Consider Non-Trivial Cases | ||
| S Oct 12, 2020 at 20:21 | history | suggested | xion3582 | CC BY-SA 4.0 |
changed the language so that it doesn't assume the gender of the reader
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| Oct 12, 2020 at 18:52 | review | Suggested edits | |||
| S Oct 12, 2020 at 20:21 | |||||
| Oct 12, 2020 at 16:58 | comment | added | Consider Non-Trivial Cases | @WillJagy that was very helpful, this time I was looking for a general solution. | |
| Oct 12, 2020 at 16:41 | comment | added | Will Jagy | @Servaes first posted at MSE math.stackexchange.com/questions/3820129/… | |
| Oct 12, 2020 at 7:38 | comment | added | user75451 | This is really not a question for MO, but for MSE. | |
| Oct 12, 2020 at 7:10 | history | became hot network question | |||
| Oct 12, 2020 at 7:02 | comment | added | GH from MO | @Andrew: Every prime $p\equiv 2\pmod{3}$ is represented by $x^2+y^2$ or $3x^2-y^2$. See my response for more detail. | |
| Oct 12, 2020 at 6:43 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Oct 12, 2020 at 6:08 | history | edited | Consider Non-Trivial Cases | CC BY-SA 4.0 |
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| Oct 12, 2020 at 6:04 | answer | added | GH from MO | timeline score: 14 | |
| Oct 12, 2020 at 4:38 | comment | added | Consider Non-Trivial Cases | @GerryMyerson you are correct, then what is the solution and where I can find related results? | |
| Oct 12, 2020 at 4:36 | comment | added | Consider Non-Trivial Cases | @WillSawin yes overlap is allowed, then, for example, how we find quadratic forms that represents all primes defined by $p \equiv 2 \pmod 3$? | |
| Oct 12, 2020 at 4:31 | history | edited | Consider Non-Trivial Cases | CC BY-SA 4.0 |
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| Oct 12, 2020 at 4:02 | comment | added | Gerry Myerson | I read the question as permitting overlap. I think the question is, is there a finite set of (preferably irreducible) binary quadratic forms such that every prime is represented by at least one of the forms in the set? | |
| Oct 11, 2020 at 23:37 | comment | added | Will Sawin | You can certainly get all primes if you are willing to accept overlap between the different quadratic forms. | |
| Oct 11, 2020 at 23:25 | comment | added | Will Sawin | If a quadratic form represents $p$, then the discriminant $b^2 -4ac$ of the form is a perfect square modulo $p$. (Proof: The discriminant is invariant under change of variables. Because the form represents $p$, we can change variables so that $a=p$.) So indeed no non-split form represents all primes and sets like the set of primes congruent to $2$ mod $3$ can never be represented. | |
| Oct 11, 2020 at 23:11 | comment | added | Wojowu | Actually on second reading, you didn't ask for the form to only represent the forms of specific residue. In that case there are some trivial examples, like the form $xy$, or $x^2-y^2$. I'm not sure if you can do with only irreducible forms. | |
| Oct 11, 2020 at 23:08 | comment | added | Wojowu | I'm fairly sure that for any primitive quadratic form and any $m>0$, the form represents infinitely many primes which are $1\pmod m$. This should follow from Chebotarev's density theorem. Therefore there cannot be such a form which only represents non-1 residues modulo a number. | |
| Oct 11, 2020 at 22:56 | history | asked | Consider Non-Trivial Cases | CC BY-SA 4.0 |