Timeline for Integers $d$ for which the negative Pell equation is soluble for both $d$ and $2d$?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 30 at 14:25 | comment | added | user967210 | I added a new condition, but I don't know how to prove it | |
| Sep 30 at 14:25 | history | edited | user967210 | CC BY-SA 4.0 |
Added a new condition
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| Sep 30 at 11:53 | comment | added | user967210 | @TitoPiezasIII Sorry, I had to add that further condition, as OscarLanzi explained | |
| Sep 28 at 13:43 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
added 13 characters in body
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| Sep 28 at 9:59 | comment | added | Oscar Lanzi | I looked at the reference and it adds the condition that $2^{(p−1)/4}\equiv−1\bmod p$. Thus $2^{10}\equiv−1\bmod41$ but $2^{18}\equiv+1\bmod73$, and solvability of the negative Pell equation follows suit. | |
| Sep 28 at 9:31 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
Correction made.
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| Sep 28 at 5:03 | comment | added | Tito Piezas III | @user967210 I just noticed that regarding prime $p \equiv 9\;\rm mod\;16$ for $x^2-2py^2=-1$, while $p=41$ is solvable, $p=16\cdot4+9=73$ and $p=16\cdot5+9=89$ are not, contrary to your last sentence. Care to explain? | |
| Jun 22 at 18:57 | comment | added | Will Jagy | It appears Whitford skips over that one. However, on page 80 he lists conditions 1,2,3,4, but the refers to conditions 3,4,5 . Perhaps he meant to include a condition 5 with the simple $(p,q) = -1,$ or meant that as condition 4 and the one with quartic residues moved to condition 5. | |
| Jun 22 at 18:36 | comment | added | Will Jagy | Note that Mordell's proof can be easily adapted to this: primes $p,q \equiv 1 \pmod 4$ but mutual quadratic non-residues, that is Legendre symbol $(p,q) = -1$ then there are integer solutions to $x^2 - pq y^2 = -1$ The first two semiprimes where impossibility is a surprise are $205 = 5 \cdot 41$ and $221 = 13 \cdot 17$ both times mutual residues. | |
| Mar 25, 2022 at 17:15 | history | edited | Glorfindel | CC BY-SA 4.0 |
formatting
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| S Mar 25, 2022 at 16:31 | review | First answers | |||
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| S Mar 25, 2022 at 16:31 | history | edited | user967210 | CC BY-SA 4.0 |
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| Mar 25, 2022 at 15:15 | review | Late answers | |||
| Mar 25, 2022 at 15:24 | |||||
| S Mar 25, 2022 at 14:54 | review | First answers | |||
| Mar 25, 2022 at 15:24 | |||||
| S Mar 25, 2022 at 14:54 | history | answered | user967210 | CC BY-SA 4.0 |