Timeline for Can you find squares in this class?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 20, 2015 at 16:28 | vote | accept | OriginalBBB | ||
| Mar 19, 2015 at 12:40 | answer | added | Jeremy Rouse | timeline score: 6 | |
| Mar 18, 2015 at 17:04 | comment | added | OriginalBBB | @JeremyRouse That is great news for me (and definitevely an answer to my question - you should add it as an answer), could you please point me to some reference about the informations you used in your deduction? | |
| Mar 18, 2015 at 16:52 | comment | added | Jeremy Rouse | If there is a solution, then $p \equiv 1 \pmod{12}$. You are asking about rational points on the hyperelliptic curve $y^{2} = p(x^{4} + 6x^{2} - 3)$. There are no $3$-adic solutions if $p \equiv 2 \pmod{3}$, and there are no $p$-adic solutions unless $p \equiv \pm 1 \pmod{12}$. It follows that all solutions have $p \equiv 1 \pmod{12}$. There are also solutions if $p = 193, 349, 373, 433, 601, ...$. | |
| Mar 18, 2015 at 16:00 | history | edited | OriginalBBB | CC BY-SA 3.0 |
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| Mar 18, 2015 at 15:58 | comment | added | OriginalBBB | Thanks @jmc, are there no solutions between the primes 37 and 8929? Futhermore, all the primes you found are congruent to 1 mod 12, so the class of primes congruent to 7 mod (12) seems to be outside of it (a guilible mathematician would say). Any ideas? | |
| Mar 18, 2015 at 15:27 | comment | added | jmc | And more tuples $(l,m,p)$: (7, 8, 8929) (8, 11, 6637) (10, 9, 38917) (11, 8, 48817) (13, 10, 99961) (14, 9, 113989) (16, 7, 133597) (17, 6, 142057) | |
| Mar 18, 2015 at 15:23 | comment | added | jmc | Taking $(l,m) = (8,1)$ gives $11^{2} \cdot 37$. So combine that with $p = 37$, and you get a square. | |
| Mar 18, 2015 at 15:21 | comment | added | jmc | I quickly tested whether $l^{4} + 6l^{2}m^{2} - 3m^{4}$ is square-free for $l,m \in \{1,\ldots,50\}$, and there are 1491 tuples where it is not. | |
| Mar 18, 2015 at 14:40 | history | edited | OriginalBBB | CC BY-SA 3.0 |
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| Mar 18, 2015 at 14:25 | history | edited | OriginalBBB | CC BY-SA 3.0 |
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| Mar 18, 2015 at 14:24 | review | First posts | |||
| Mar 18, 2015 at 14:29 | |||||
| Mar 18, 2015 at 14:19 | history | asked | OriginalBBB | CC BY-SA 3.0 |