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Mar 16, 2022 at 21:22 history edited Yuan Yang CC BY-SA 4.0
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Mar 16, 2022 at 13:47 history edited Yuan Yang CC BY-SA 4.0
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Mar 16, 2022 at 12:13 history edited Yuan Yang CC BY-SA 4.0
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Mar 16, 2022 at 12:12 comment added Yuan Yang @JoachimKönig You are so right, thank you so much for pointing out!
Mar 16, 2022 at 12:10 comment added Joachim König @YuanYang With the corrected coefficients, Magma says $r=1/8$ is a counterexample.
Mar 16, 2022 at 12:01 comment added Will Sawin @StanleyYaoXiao Couldn't a disproof consist of a single counerexample? I agree proving the proportion where at least one has rank $0$ is exactly $75\%$ is beyond current technology.
Mar 16, 2022 at 12:00 comment added Gerry Myerson This is D19 in Guy, Unsolved Problems In Number Theory, where there is a discussion and loads of references to the literature. Worth a look, if you are interested in the problem and haven't already seen Guy's book.
Mar 16, 2022 at 11:57 comment added Yuan Yang @JoachimKönig Ahhh thank you for pointing out! I had a typo it should be $-2/r^2 x+1/r^2$
Mar 16, 2022 at 11:56 history edited Yuan Yang CC BY-SA 4.0
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Mar 16, 2022 at 11:54 comment added Joachim König @YuanYang: No, I mean the way the constant coefficient is currently written as $-2/r^2+1/r^2$, it looks like a typo (and also computation doesn't show what you claim with this form).
Mar 16, 2022 at 11:53 comment added Chris Wuthrich Related : mathoverflow.net/questions/301991/four-distance-problem
Mar 16, 2022 at 11:51 comment added Yuan Yang @JoachimKönig when $r=0$ it’s a degenerate case; the relation of $r$ and $1-r$ is crucial so I stayed this form
Mar 16, 2022 at 11:45 comment added Joachim König Is there a mistake in the constant coefficient of your family (or possibly $x$ missing for the linear term) ?
Mar 16, 2022 at 11:40 comment added Stanley Yao Xiao Given general conjectures about ranks of elliptic curves your conjecture is likely false. The amount of computational evidence you provided to support your conjecture is paltry at best. Regardless, either a proof or disproof of your conjecture for all $r$ is likely beyond current technology.
Mar 16, 2022 at 11:25 history asked Yuan Yang CC BY-SA 4.0