Timeline for Solve this Diophantine equation $(2^x-1)(3^y-1)=2z^2$
Current License: CC BY-SA 4.0
26 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S May 14, 2021 at 5:02 | history | bounty ended | CommunityBot | ||
| S May 14, 2021 at 5:02 | history | notice removed | CommunityBot | ||
| May 9, 2021 at 1:18 | comment | added | Tomita | Assuming $x=1.$ According to Silverman's comment, we get three elliptic curves as follows. $ y=3w \implies Y^2=X^3-8$ with integral points=$(2,0)$, $ y=3w+1\implies Y^2=X^3-72$ with integral points=$(6,\pm12)$, $ y=3w+2\implies Y^2=X^3-648$ with integral points=$(9,\pm 9), (18,\pm 72), (22,\pm 100), (54,\pm 396), (97,\pm 955), (1809,\pm 76941).$ | |
| May 8, 2021 at 21:40 | answer | added | Derivative | timeline score: 4 | |
| May 6, 2021 at 6:22 | history | edited | YCor | CC BY-SA 4.0 |
added tag, fixed typo, formatting
|
| S May 6, 2021 at 3:48 | history | bounty started | math110 | ||
| S May 6, 2021 at 3:48 | history | notice added | math110 | Authoritative reference needed | |
| S Nov 21, 2019 at 9:00 | history | bounty ended | CommunityBot | ||
| S Nov 21, 2019 at 9:00 | history | notice removed | CommunityBot | ||
| S Nov 13, 2019 at 7:20 | history | bounty started | math110 | ||
| S Nov 13, 2019 at 7:20 | history | notice added | math110 | Authoritative reference needed | |
| Aug 13, 2018 at 18:55 | comment | added | Kieren MacMillan | Is the [possible] connection to Fermat’s “elliptic curve” (Y^2+4=X^3) coincidental? | |
| S Apr 19, 2018 at 4:29 | history | bounty ended | CommunityBot | ||
| S Apr 19, 2018 at 4:29 | history | notice removed | CommunityBot | ||
| Apr 17, 2018 at 1:52 | comment | added | j.c. | Ah, I was assuming from @GerhardPaseman's comment that there was an easy proof that I was missing... | |
| Apr 16, 2018 at 22:08 | comment | added | Noam D. Elkies | Thanks, but that's still just the $x=1$ case so it wouldn't qualify as an answer to the bounty question. | |
| Apr 16, 2018 at 19:49 | comment | added | j.c. | Following the hint of @NoamD.Elkies , I found a 1958 paper of CY Lee "Some properties of nonbinary error-correcting codes" ieeexplore.ieee.org/document/1057446 which attributes the solution of $2z^2+1=3^y$ to T. Nagell. Since it is hard to find, I have uploaded a copy of Nagell's 1923 paper "Sur l'impossibilité de quelques equations à deux indéterminées" here dropbox.com/s/69iuu00ux30dclu/nagell1923.pdf?dl=0 (this dropbox link is not really permanent, so rehosting would be appreciated). Perhaps someone more expert than me could write an answer? | |
| Apr 11, 2018 at 2:49 | comment | added | Noam D. Elkies | The Diophantine equation $2z^2+1 = 3^y$ that appears for $x=1$ has already been studied: $2z^2+1 = 4{z \choose 2} + 2z + 1$ is the volume of a radius-2 ball in ternary Hamming space ${\bf F}_3^z$, so $2z^2+1 = 3^y$ is a necessary condition for the existence of a perfect 2-error-correcting ternary code in that space. It is known that $z=1,2,11$ are the only solutions, each corresponding to a code (trivial codes for $z=1$ and $2$, and the ternary Golay code for $z=11$). | |
| S Apr 11, 2018 at 2:30 | history | bounty started | math110 | ||
| S Apr 11, 2018 at 2:30 | history | notice added | math110 | Authoritative reference needed | |
| Apr 1, 2018 at 18:14 | comment | added | Igor Rivin | Actually, the argument in the linked-to reference goes through essentially line by line, with nary an elliptic curve. | |
| Apr 1, 2018 at 12:26 | comment | added | Joe Silverman | And after you show that $x=1$, as Gerhard suggests, you can take the three cases $y=3w$, $y=3w+1$, and $y=3w+2$, which reduces the problem to finding the integer points on three fairly standard elliptic curves. | |
| Apr 1, 2018 at 11:49 | history | edited | math110 | CC BY-SA 3.0 |
added 1 character in body
|
| Apr 1, 2018 at 5:33 | comment | added | Gerhard Paseman | Can you show that x has to be one? Gerhard "Try It. It's Not Hard" Paseman, 2018.03.31. | |
| Apr 1, 2018 at 4:56 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
minor typos
|
| Apr 1, 2018 at 4:47 | history | asked | math110 | CC BY-SA 3.0 |