Timeline for Estimating the size of solutions of a diophantine equation
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
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| Jun 3 at 17:21 | comment | added | Rosie F | Let $x=b+c, y=a+c, z=a+b$. Does your result extend to $x>0, y>0, z>0$ even if $a<0$? Your equation is equivalent to $(x+y+z)(1/x+1/y+1/z)=N [1]$ where $N=2n+6$, so the extension would prove that [1] has no solution when $4\mid N$ and $N\geqslant 8$. [1] is discussed in Andrew Bremner, Richard Guy, Richard Nowakowski. Which integers are representable as the product of the sum of three integers and the sum of their reciprocals? Mathematics of Computation, 61, no. 203, July 1993, pp. 117--130. | |
| Mar 26, 2017 at 14:17 | comment | added | Paul-Olivier Dehaye | I got here through a Facebook nerd snipe: i.sstatic.net/hocjV.jpg | |
| Mar 26, 2017 at 14:16 | comment | added | Paul-Olivier Dehaye | For those it helps, the curve is documented in the LMFDB: lmfdb.org/EllipticCurve/Q/910/a/4 | |
| Mar 25, 2017 at 20:14 | history | edited | GH from MO | CC BY-SA 3.0 |
fixed a typo in the second case of the "three cases"
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| Mar 25, 2017 at 19:31 | comment | added | GH from MO | Thank you for the isomorphism! I got confused, because I actually only read what is after the first horizontal line, thinking that the old version was above it :-) | |
| Mar 25, 2017 at 18:14 | comment | added | Michael Stoll | @GHfromMO (continued) In the concrete case, an isomorphism is given by setting $x = 4(n+3)(a+b+2c)/(c-(n+2)(a+b))$ and $y = (8n^2+44n+60)(a-b)/(c-(n+2)(a+b))$ (according to Magma). | |
| Mar 25, 2017 at 18:11 | comment | added | Michael Stoll | @GHfromMO You should actually stop reading at the first horizontal line; what is below are my somewhat incremental thoughts leading to the solution that is spelled out in some detail above the line. To get the isomorphism, one changes coordinates so that one of the inflection points is at infinity and the tangent at that points is the line at infinity (this works here, since there is a rational inflection point; in general, there is still an isomorphism with a curve in Weierstrass form, but it is more complicated)... | |
| Mar 25, 2017 at 16:39 | comment | added | GH from MO | Very nice indeed. Can you please add the final details, namely why $(\xi, 3-2n)_p = 1$ holds for all primes $p$? (You say that $d(n) = -5-2n$ works better, but this is not a positive integer.) Also, can you please explain why $E_n'$ is isomorphic to $E_n$? Thanks in advance! | |
| Jun 12, 2016 at 15:53 | history | edited | Michael Stoll | CC BY-SA 3.0 |
Fixed a wrong sign in the equation of $E'_n$
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| Jan 7, 2016 at 4:41 | comment | added | alex alexeq | +1 very nice. A direct proof, compared to Allan's article mentioned below. | |
| Jan 7, 2016 at 4:38 | vote | accept | alex alexeq | ||
| Jan 6, 2016 at 15:49 | history | edited | Michael Stoll | CC BY-SA 3.0 |
added several missing minus signs
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| Jan 6, 2016 at 8:55 | history | edited | Michael Stoll | CC BY-SA 3.0 |
Added clean statement and proof
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| Jan 5, 2016 at 23:26 | comment | added | R.P. | This must be one of the easiest-to-state (and most attractive!) problems in number theory whose solution "requires" the Brauer-Manin obstruction. Very cool! | |
| Jan 5, 2016 at 23:08 | history | edited | Michael Stoll | CC BY-SA 3.0 |
finished the argument
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| Jan 5, 2016 at 22:35 | history | edited | Michael Stoll | CC BY-SA 3.0 |
fixed a mistake
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| Jan 5, 2016 at 22:26 | history | edited | Michael Stoll | CC BY-SA 3.0 |
added some explanation
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| Jan 5, 2016 at 21:58 | history | edited | Michael Stoll | CC BY-SA 3.0 |
fixed a typo, improved wording
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| Jan 5, 2016 at 21:50 | history | edited | Michael Stoll | CC BY-SA 3.0 |
added 1383 characters in body
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| Jan 5, 2016 at 19:42 | history | edited | Michael Stoll | CC BY-SA 3.0 |
added 598 characters in body
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| Jan 5, 2016 at 18:44 | history | edited | Michael Stoll | CC BY-SA 3.0 |
added 235 characters in body
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| Jan 5, 2016 at 18:27 | comment | added | Michael Stoll | OK, thanks; I overlooked the positivity requirement. I'll look at larger $n$... | |
| Jan 5, 2016 at 18:20 | comment | added | Jeremy Rouse | The OP is requesting $a$, $b$ and $c$ to be positive, which appears to correspond to points on the non-identity component of the Weierstrass model of $E_{n}$. I don't think there are such points for $n = 19$. | |
| Jan 5, 2016 at 18:19 | history | answered | Michael Stoll | CC BY-SA 3.0 |
