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One can use Thue's 1909 result to show that the Diophantine equation $Ax^3 + By^3 = C$ ($A,B$ not perfect cubes, $C\neq 0$) has finitely many integer solutions $(x,y)$.

But does there exist a simple way to prove this equation (because I think this is a special equation?

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  • $\begingroup$ The simplest way, I think, is to repeat the proof by Thue in this particular case. The proof will be shorter than the general proof by Thue. $\endgroup$
    – markvs
    Feb 6, 2021 at 5:20
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    $\begingroup$ (Don’t forget about Skolem’s method!) $\endgroup$
    – alpoge
    Feb 6, 2021 at 9:27
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    $\begingroup$ (I guess I should probably tell you what that is! Here’s how I remember it. First off (ignore constants) $x^3 + ny^3 = \mathrm{Nm}(x - y \sqrt[3]{n})$, and so you’re asking if there are finitely many elements in $\mathbb{Z}[\sqrt[3]{n}]$ with no $\sqrt[3]{n^2}$ coeff. and of norm some given const. Well all the elements of norm that const. are of the form $(\text{in some explicit finite set})\cdot \eta^k$ for $\eta$ a fund. unit and $k\in \mathbb{Z}$. So go and write everything in terms of $k$, which is the only var. The sol.s correspond to the $k$ for which the $\sqrt[3]{n^2}$ coeff. is $0$. $\endgroup$
    – alpoge
    Feb 6, 2021 at 9:52
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    $\begingroup$ Your question is unanswerable because the phrase "simple proof" has no precise meaning. Please try to explain what you mean by "simple proof". Using only congruenes? Using reciprocity laws (say quadratic and cubic reciprocity)? Using reasonably elementary $p$-adic analysis such as used by Skolem? Using 110 year old Diophantine approximation methods such as those used by Thue? For an expert, all of these are relatively "simple proofs" in the sense that they use methods developed a long time ago. OTOH, for someone first studying number theory, these methods are not simple. $\endgroup$ Feb 11, 2021 at 1:35
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    $\begingroup$ I was about to write up Skolem's method but then noticed that alpoge already noted it a week ago. I might add that Skolem's method, like those of Thue et al., gives an upper bound on the number of solutions but not on their size; but unlike the case for Thue et al. one can often use Skolem's technique (sometimes with more elementary auxiliary arguments) to solve such an equation completely, by approximating each $p$-adic solution closely enough to either recognize it as an integer solution or prove that it does not come from any integer solution. $\endgroup$ Feb 13, 2021 at 5:57

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I think both of the answers given avoid the modern treatment of Thue equations, which despite being newer is probably simpler in many ways. As my doctoral advisor is a pioneer in this area I feel obligated to explain these ideas.

Let us first emphasize that Thue's theorem, extending to the theorems of Siegel, Dyson, and Roth, is completely ineffective in the height aspect (i.e., it gives no way at all to control the size of solutions) and performs very badly in the count aspect (i.e., it is in principle possible to extract a bound for the number of solutions, but it is terrible). The height of the solutions can be bounded using effective bounds on linear forms of logarithms, due to Baker. However these height bounds are terrible (super-polynomial in the coefficients of the polynomial and the integer $h$ in the equation $F(x,y) = h$) and are of little practical use. This is still the best general method for height bounds of solutions as far as I know.

In terms of bounding the number of solutions, there has been significant progress. The key idea comes from a paper of Bombieri and Schmidt: one can obtain a fairly cheap way to reduce a Thue equation $F(x,y) = h$ into a family of Thue equations $G_a(x,y) = 1$, where $a \in A$ lies in some finite indexing set. The forms $G_a$ have the same degree as $F$. One then obtains a uniform bound, depending only on the degree $d = \deg F = \deg G_a$, for the number of solutions of Thue equations of the shape $G(x,y) = 1$ which is valid for any binary form $G$ of degree $d$ with non-zero discriminant. Thus, one obtains a bound of the shape $C_d \cdot |A|$, where $C_d$ is the bound for the number of solutions to the unit equation $G(x,y) = 1$ for forms of degree $d$.

This approach exploits the fact that for any integer $m$, the set of solutions to the congruence $F(x,y) \equiv 0 \pmod{m}$ lies in a finite number of lattices of $\mathbb{Z}^2$ (that number could be 0, if the congruence is not solvable). This is because $F$ splits into a product of linear forms over an algebraically closed field. One then writes $h = p_1^{a_1} \cdots p_k^{a_k}$, and since $F(x,y) = h$ has a solution the congruence $F(x,y) \equiv 0 \pmod{p_i^{a_i}}$ is solvable for each $1 \leq i \leq k$. This implies that $F$ has a linear factor over $\mathbb{F}_{p_i}$. Each such factor gives rise to a lattice $\Lambda_j(p_i)$. For simplicity, let us assume that $F$ is unramified at $p_i$. Then we may use Hensel's lemma to show that each of the $\Lambda_j(p_i)$'s can be lifted uniquely to a lattice $\Lambda_j(p_i^{a_i})$ whose elements correspond to solutions of the congruence $F(x,y) \equiv 0 \pmod{p_i^{a_i}}$. The ramified case is a bit more delicate, but is handled in full generality by this paper of Cam Stewart.

Thus, for each $p_i^{a_i}$ we obtain a finite number of lattices (at most $d = \deg F$) $\Lambda_j(p_i^{a_i})$ and we apply a transformation sending $\mathbb{Z}^2$ to $\Lambda_j(p_i^{a_i})$, thus obtain forms $F_{\Lambda_j(p_i^{a_i})}(x,y)$. We have thus exchanged our original Thue equation $F(x,y) = h$ for at most $d$ equations of the form

$$\displaystyle F_{\Lambda_j(p_i^{a_i})}(x,y) = h p_i^{-a_i}.$$

We may then repeat this process, and eventually obtain a finite number $A$ (roughly $d^k = d^{\omega(h)}$ many) of lattices and auxiliary forms $G_a$ with $a \leq A$, and the unit equation $G_a(x,y) = 1$. Assuming we can obtain a bound for these latter equations which depends only on $d$, then we will get a bound roughly of the shape $C_d \cdot d^{\omega(h)}$, which is $O_{d,\epsilon}(h^\epsilon)$ for any $\epsilon > 0$.

How are such uniform bounds achieved? The key observation is that binary forms with integer coefficients and non-zero discriminant has discriminant at least one in absolute value. In particular, we are now in a situation where the discriminant is comparably large compared to the integer $h$. This means that we can rely on the Mahler measure of the form (really the corresponding polynomial $F(x,1)$) and the discriminant alone to bound the number of large solutions, using the Thue-Siegel principle, which is an effective (and very simple, i.e., does not require the construction of auxiliary polynomials as in Thue's method) but fairly weak way to push apart large solutions. One then bounds the small solutions suitably, and then optimize the counting by controlling various parameters.

I should mention that Stewart makes a critical observation in his 1991 JAMS paper linked above, where he exploits this principle further by noting that it is really the property that $h$ is small compared to the discriminant of $F$ that matters. Here one observes that as one replaces the forms $F$ with the forms $F_{\Lambda_j(p_i^{a_i})}$ the discriminant actually increases substantially, thus one can often move into a favourable situation where one obtains a finite number of equations of the shape $G_a(x,y) = g$ with $g$ very small (but not necessarily equal to one) compared to the discriminant of $G_a$ without using all of the prime factors of $h$.

This is still essentially the best known result, despite the passage of nearly 30 years (since Stewart's 1991 paper). Indeed, it is conjectured that there should be a uniform bound depending only on the degree $d$ such that the number of solutions to the Thue equation $F(x,y) = h$ is bounded by $O_d(1)$. This is a special case of the uniform boundedness conjecture, which is known to be true assuming the Bombieri-Lang conjecture (this result is due to Caporaso, Harris, and Mazur) Stewart makes an even stronger conjecture in his aforementioned paper.

Indeed, the Bombieri-Schmidt theorem gives unconditionally the following special case of the uniform boundedness conjecture, which I don't believe can be proved any other way: we consider the family $\mathcal{C}(d)$ of affine curves defined by two objects: a binary form $F$ with non-zero discriminant, integer coefficients, and degree $d$, and a prime number $p$ - then the affine curves

$$C(F,p) : = \{F(x,y) = p\}$$

have a uniformly bounded number of integral points (i.e., primitive solutions to the Thue equation $F(x,y) = p$). Indeed, one can get something very explicit... for instance $2800 d^2$ works. This result does not require any input from the Mordell-Weil rank of the Jacobian of the curve, or any other similar quantity (i.e., Minhyong Kim's notion of global Selmer variety).

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It really depends on what you mean by the word "simple" in "simple proof". Any proof that your cubic equation always has finitely many integer solutions will imply that if $D\in\mathbb Z$ is not a perfect cube, then for every $\epsilon>0$ there are only finitely many rational numbers $p/q\in\mathbb Q$ satisfying $$ \left| \frac{p}{q}-\sqrt[3]{D} \right| \le \frac{1}{q^{3+\epsilon}}. $$ This is a moderately deep theorem whose known proofs involve either the auxiliary polynomial methods used in Diophantine approximation and transcendence theory or $p$-adic analytic methods (as noted in other comments/answers, these methods that are associated to the names Thue/Siegel/Roth ... and Skolem ...).

In particular, if by "simple proof" you mean something along the lines of $x^2+3y^2=z^2$ has no non-trivial solutions in integers because we can reduce mod $3$, or $y^2=x^3+7$ has no solutions in integers because (with a little bit of algebra) we can reduce it to a question about squares mod $q$ and use quadratic reciprocity, then the answer to your question is no. There are no proofs that I am aware of that use only, say, congruence considerations and cubic reciprocity.

On the other hand, as has been mentioned in other comments and answers, carrying out Thue's argument for $Ax^3+By^3=C$ is somewhat easier than the general case. But I'm not sure it could be considered a "simple proof."

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  • $\begingroup$ I think Mahler's proof that for any positive number $M$ and any irreducible binary form $F$ of degree $d \geq 3$ the inequality $|F(x,y)| \leq M$ has at most finitely many solutions in integers $x,y$ does not involve constructing auxiliary polynomials as in the proofs of Thue-Siegel-Roth, so in principle should be considered simpler. Mahler's result would of course imply finiteness to the number of solutions to any particular Thue equation $\endgroup$ Feb 6, 2021 at 19:00
  • $\begingroup$ @StanleyYaoXiao Interesting. How does Mahler prove that result without an a priori Diophantine approximation estimate? $\endgroup$ Feb 6, 2021 at 19:09
  • $\begingroup$ OK so I’m going to go out on a bit of a limb here because I regrettably can’t read German (though hopefully that’ll change soon), so I could well be contributing garbage. Nonetheless, it would be quite a surprise for Mahler to have an argument not using auxiliary polynomials etc., so I tried to find the relevant paper / theorem. It seems the relevant reference is Mahler’s “Zur Approximation Algebraischer Zahlen III”, with the main theorem stated in the introduction. If I’m not mistaken (and this matches with the natural strategy for him to have used), the key input is Satz I of that paper. $\endgroup$
    – alpoge
    Feb 7, 2021 at 10:56
  • $\begingroup$ Now he doesn’t give a pf. of Satz I, but rather (again, I think, but I can’t be sure because I can’t actually read it) ref.s paper I in the series, i.e. “Zur Approx. Algebraisch. Zahlen I” —- specifically, Satz I on p. 710 of that paper. Well, that paper seems to be full of the expected techniques (following Siegel, I think) —- obv. since I can’t read it I can’t conclude with certainty, but the fact that he states a bound with Siegel’s exponent (+ the notation & the word “Wronskian”) is what led me to think it’s of the usual “poly. method” type. But please do let me know if I’ve guessed wrong! $\endgroup$
    – alpoge
    Feb 7, 2021 at 11:05
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    $\begingroup$ @alpoge (the above comment was also for you, but I could only @ one person at a time) I also cannot read German, but my younger brother Anton translated Mahler's paper while we were both students of Cam Stewart. I think the modern consensus is that Mahler's ideas from the 30s allows one to avoid the fundamental ineffectivity of Thue's original approach (extending to Roth's theorem), and is the foundation of modern effective bounds on the number of solutions to Thue equations. As far as I know only the cardinality of solutions is effective, not the height $\endgroup$ Feb 11, 2021 at 1:55
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It can be reduced to Mordell equation: $$Y^2 = X^3 + (4ABC)^2$$ with $Y:=4AB(2By^3-C)$ and $X:=-4ABxy$, which was shown by Mordell to have finitely many integer solutions.

ADDED. M. A. Bennett and A. Ghadermarzi (2015) explored the connection between Mordell equations and cubic Thue equations, and computed all solutions for the former with the free terms below $10^7$ by absolute value.

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    $\begingroup$ That's about as difficult as using Thue's theorem, no? $\endgroup$
    – JoshuaZ
    Feb 6, 2021 at 2:21
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    $\begingroup$ It is much harder than Thue's proof. $\endgroup$
    – markvs
    Feb 6, 2021 at 2:47
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    $\begingroup$ Mordell reduced it (= finiteness of integral points on elliptic curves over $\mathbb{Q}$) to Thue, and Siegel’s more general proof reduces it (= finiteness of integral points on genus $g$ curves, subject to the usual conditions about the divisor at infinity when $g\leq 1$) to Roth (well actually to Siegel’s work, I believe from his undergraduate years / ultimately his thesis, improving Thue’s bound). $\endgroup$
    – alpoge
    Feb 6, 2021 at 9:31

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