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The Wikipedia (https://en.wikipedia.org/wiki/1000_(number)#1300_to_1399) mentions that 1331 is the only cube of the form $x^2 + x − 1$, for $x = 36$. What is the proof?

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    $\begingroup$ let me swap your variables and try to solve y^2 + y = x^3 + 1. That looks like an elliptic curve, and in fact it's this one: lmfdb.org/EllipticCurve/Q/225/c/2 In fact what you write is not quite true, since 0^2 + 0 - 1 = (-1)^3, and 1^2 + 1 - 1 = 1^3. But the solution you've got is sort of the interesting one. $\endgroup$
    – pupshaw
    Jun 9, 2021 at 18:42
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    $\begingroup$ @NoahSchweber Why? There is non-trivial mathematics here, and the OP knows her mathematics $\endgroup$
    – Yemon Choi
    Jun 9, 2021 at 20:05
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    $\begingroup$ To correct Thomas Browning's comment above, the curve is actually rank $1$, so there are infinitely many rational solutions, stating $(61/25, 434/125)$, $(-71/144, 973/1728)$, $(-779/1849, -125945/79507)$ ... $\endgroup$ Jun 10, 2021 at 0:46
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    $\begingroup$ According to LMFDB, $y^2+y=x^3+1$ has integral solutions as follows. $ (x,y)=(-1, 0) , (-1, -1) , (1, 1) , (1, -2) , (11, 36) , (11, -37). $ Hence only positive integral solutions are $(x,y)=(1, 1),(11, 36).$ Using generator $(x,y)=(1,1)$, we can get positive rational solutions such as $(61/25, 434/125),(408241/45796, 256170307/9800344),(479399/2739025, 2812459393/4533086375)$. $\endgroup$
    – Tomita
    Jun 10, 2021 at 3:47
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    $\begingroup$ ... The Dipohantine equation $u^2-v^3=k$ is known as *Mordell's equation". It has been extensively studied; in particular, Mordell has shown that for any integer $k$ the equation has only a finite number of solutions. For some values of $k$, Mordell's equation can be solved using an elementary argument, not appealing to elliptic curves. I do not know whether the case $k=80$ admits an elementary solution, just search the web / arXiv. $\endgroup$
    – Seva
    Jun 11, 2021 at 8:14

2 Answers 2

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Here is the more elementary way that one would have solved this before LLL. I will still use the computer for some calculations, but technically one could do all this by hand. Worse, I won't even do all of the steps, but I hope the part I show explains the method, for those that may be interested. Maybe someone wants to complete it or finds a shorter way.

First, setting $K=\mathbb{Q}(\theta)$ with $\theta^2+\theta-1$, a.k.a the Golden ratio, we will rewrite our equation as $$N_{K/\mathbb{Q}}(y-\theta) = x^3$$ to be solved in $X$ and $y$ in $\mathbb{Z}$. The field $K$ has class number 1 and the units are generated by $\theta$. From this, using a little argument about the $\sqrt{5}$-adic valuation, one can show that, there must be a $i\in \{0,1,2\}$ such that $$y-\theta = \theta^i\cdot \alpha^3$$ for some $\alpha\in\mathbb{Z}[\theta]$, the ring of integers of $K$. Write $\alpha = a+b\,\theta$ for $a,b\in\mathbb{Z}$ and split into the three cases according to $i$.

Case $i=0$ : Spanning out $\alpha^3$ and equating the coefficients in front of $\theta$, we get $$\begin{align*} y &= a^3+3ab^2-b^3 \\ -1&=b\cdot(3a^2-3ab+2b^2)\end{align*}$$ Now the second line tells us that $b=\pm 1$, but neither of the choices allows us to find $a$ as an integer, so that case does not occur.

Case $i=1$ : This time the equations are $$\begin{align*} y &= 3a^2b-3ab^2+2b^3\\-1 &=a^3-3a^2b+6ab^2-3b^3\end{align*}$$ One can note here aside that the second line is a model of the curve $3$-isogenous to the original curve over $\mathbb{Q}$. This second (Thue) equation, we rewrite as $$-1 = N_{L/\mathbb{Q}}(a-b\xi)$$ where $L=\mathbb{Q}(\xi)$ and $\xi^3-3\xi^2+6\xi-3=0$. This $L$ has also trivial class number and the units are generated by $1-\xi$ of norm $1$. We are now looking for units of norm $-1$ in $L$ with no $\xi^2$ term. Once more we split into three cases $j\in\{0,1,2\}$. We want to solve $$a+b\xi = - (1-\xi)^j\cdot(1-\xi)^{3k} = -(1-\xi)^j\cdot(-2+3\xi)^k.$$

Subcase $j=0$ : We may rewrite it as $$a-b\xi = - (1+3(\xi-1))^k = -\bigl(1+3(\xi-1)k+9(\xi-1)^2k(k-1)/2 + 27\cdots \bigr)$$ and consider this as a power series in $k$ over $\mathbb{Q}_3(\xi)$, which is a totally ramified extension of $\mathbb{Q}_3$. We can compare the power series coefficient in front of $\xi^2$, to get an equation of the form $$0=9 k (k-1)\cdot \bigl(1/2 -9/24 (k-2)(k-3)+\cdots\bigr).$$ By Strassmann's theorem, the power series has at most two solutions in $\mathbb{Q}_3$. But it is easy to spot that there are two such solutions, namely $(a=-1,b=0,k=0)$ and $(a=2,b=3,k=1)$. These correspond to $(x,y)=(-1,0)$ and $(11,36)$ repsectively.

Subcase $j=1$ : leads to a single solution $a-b\xi=-1+\xi$ which corresponds to $(x,y)=(1,-2)$.

Subcase $j=2$ : produces a unit power series and has therefore no solutions.

Case $i=2$ : I believe one just gets the $-P$ for all the $P$ found in the case $i=1$. But I admit that I have not checked.

Finally, I should say that for questions on integral points on elliptic curve I often turn to Smart's "The algorithmic Resolution of Diophantine Equations" that explains things well. Like the $3$-adic method used in the subcases in chapter III.

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    $\begingroup$ Regarding the $i=2$ case: Given a solution to $y-\theta = \theta^2 \alpha^3$, rewrite it as $y - \theta = \theta^{-1} \beta^3$ for $\beta = \theta \alpha$. Then apply the Galois automorphism of $K$ to get $y-(1-\theta) = - \theta \beta^3$ or $(1-y) - \theta = \theta \beta^3$. So, if $y$ solves the $i=2$ case then $1-y$ solves the $i=1$ case (and vice versa). And thanks for this answer! $\endgroup$ Jun 11, 2021 at 12:29
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According to SAGE, there are 6 integral points on the curve.

sage: EllipticCurve([0,0,1,0,1]).integral_points(both_signs=True)
[(-1 : -1 : 1),(-1 : 0 : 1),(1 : -2 : 1),(1 : 1 : 1),(11 : -37 : 1),(11 : 36 : 1)]
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    $\begingroup$ The question is “What is the proof?” — so “Because the computer says so” seems a slightly unsatisfying answer, on its own. It would be greatly improved by some information about how SAGE computes this, since that’d provide at least a recipe for the proof. In 5 minutes of Googling I wasn’t easily able to work out what algorithm is used — the best I could find was the SAGE documentation and a couple of old MO answers: mathoverflow.net/a/46778/2273, mathoverflow.net/a/42044/2273 $\endgroup$ Jun 10, 2021 at 15:13
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    $\begingroup$ @PeterLeFanuLumsdaine: I agree. It is more than nothing though, and the algorithms of SAGE are pretty reliable. $\endgroup$
    – GH from MO
    Jun 10, 2021 at 15:50
  • $\begingroup$ Oh yes, I completely agree this answer is an excellent starting point! $\endgroup$ Jun 10, 2021 at 15:54
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    $\begingroup$ See also How to compute rational or integer points on elliptic curves. $\endgroup$ Jun 10, 2021 at 16:05

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