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The 4-distance problem is an open question(as far as I know it is still open) that asks if there exists a point P on the Euclidean plane such that its distances to all four points of a unit square are rational numbers.

I have related this problem(see here) to questions about a family of elliptic curves $E_r$, parametrized by a rational number $r$. One can easily prove that as long as for any $r$, if one of the curves $E_r$ and $E_{1-r}$ has rank $0$, then the 4-distance problem has a negative answer, i.e. there is no such point exists.(however this is not true according to Joachim‘s comment.)

Roughly speaking, if the unit square is $A=(0,0),B=(0,1),C=(1,0),D=(1,1)$, then $E_r$ is a equation for points $P=(x,y)$ such that $x=r$ and $PA$ and $PB$ are rational numbers.

The family $E_r$ is represented by $y^2=x^3+(\frac{1}{r^2}-1)x^2-\frac{2}{r^2}x+\frac{1}{r^2}$. For $r=0, \frac{1}{2}, \frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{2}{5}$, I have verified that at least one of the curve is rank $0$. So I wonder for what set of rational numbers the rank of $E_r$ is greater than 0?

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  • $\begingroup$ 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. $\endgroup$
    – Stanley Yao Xiao
    Commented Mar 16, 2022 at 11:40
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    $\begingroup$ Is there a mistake in the constant coefficient of your family (or possibly $x$ missing for the linear term) ? $\endgroup$
    – Joachim König
    Commented Mar 16, 2022 at 11:45
  • $\begingroup$ @JoachimKönig when $r=0$ it’s a degenerate case; the relation of $r$ and $1-r$ is crucial so I stayed this form $\endgroup$
    – Yuan Yang
    Commented Mar 16, 2022 at 11:51
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    $\begingroup$ 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. $\endgroup$
    – Gerry Myerson
    Commented Mar 16, 2022 at 12:00
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    $\begingroup$ @YuanYang With the corrected coefficients, Magma says $r=1/8$ is a counterexample. $\endgroup$
    – Joachim König
    Commented Mar 16, 2022 at 12:10

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