On the practical solution of the Thue equation.

*(English)*Zbl 0657.10014This article is a break-through. - Every one working in the field of diophantine equations has the feeling that if there are integer solutions these must be “small” (and perhaps in this context “small” should be interpreted as in the range of todays or next two generations computers).

In the case of the equation \(f(X,Y)=m\) (f\(\in {\mathbb{Z}}[X,Y]\) an irreducible binary form of degree at least 3, and m a given nonzero rational integer) Thue has proved that there are only finitely many integer solutions x,y. Baker’s method gives an astronomical high upper bound for these solutions.

The authors consider four cases for Thue’s equation: the “very small”, “small”, “large”, and “very large” ones. They are able, considering the newest work in this field - almost all in preprint form -, to exclude the last (Baker’s) case. They give a very involved reduction algorithm to convert possible “large” solutions into “small” ones (at least logarithmically smaller than the initial ones) to detect that there are no “large” solutions.

A special example is given for reducing a certain upper bound A of the size of \(10^{40}\) by a computational reduction technique based on diophantine approximation theory to \(A\leq 72\). Applying the same method again gives \(A\leq 10.\)

For “small” solutions there is a continued fraction algorithm computing the solutions. “Very small” solutions must be detected by hand.

Implicitly assumed in all these considerations is that the elementary units of the algebraic number field given by the zeros of \(f(X,1)=0\) are known, equally well as the prime factorization of m in this field.

The article concludes with applying the theory of the diophantine equation \(y^ 2=x^ 3-4x+1\) where a complete solution was given by S. P. Mohanty [ibid. 30, No.1, 86-93 (1988; Zbl 0651.10011)] whose proof A. Bremner has claimed to be faulty (there are some implications of type “d \(| ab\) \(\Rightarrow\) d \(| a\) or d \(| b'')\). They arrive from this elliptic curve at a couple of totally real quartic Thue equations which are suited to their method.

This article is in some sense highly technical and the reviewer is not able to check the correctness of all the theorems. Nevertheless reading this article is a much rewarding experience.

In the case of the equation \(f(X,Y)=m\) (f\(\in {\mathbb{Z}}[X,Y]\) an irreducible binary form of degree at least 3, and m a given nonzero rational integer) Thue has proved that there are only finitely many integer solutions x,y. Baker’s method gives an astronomical high upper bound for these solutions.

The authors consider four cases for Thue’s equation: the “very small”, “small”, “large”, and “very large” ones. They are able, considering the newest work in this field - almost all in preprint form -, to exclude the last (Baker’s) case. They give a very involved reduction algorithm to convert possible “large” solutions into “small” ones (at least logarithmically smaller than the initial ones) to detect that there are no “large” solutions.

A special example is given for reducing a certain upper bound A of the size of \(10^{40}\) by a computational reduction technique based on diophantine approximation theory to \(A\leq 72\). Applying the same method again gives \(A\leq 10.\)

For “small” solutions there is a continued fraction algorithm computing the solutions. “Very small” solutions must be detected by hand.

Implicitly assumed in all these considerations is that the elementary units of the algebraic number field given by the zeros of \(f(X,1)=0\) are known, equally well as the prime factorization of m in this field.

The article concludes with applying the theory of the diophantine equation \(y^ 2=x^ 3-4x+1\) where a complete solution was given by S. P. Mohanty [ibid. 30, No.1, 86-93 (1988; Zbl 0651.10011)] whose proof A. Bremner has claimed to be faulty (there are some implications of type “d \(| ab\) \(\Rightarrow\) d \(| a\) or d \(| b'')\). They arrive from this elliptic curve at a couple of totally real quartic Thue equations which are suited to their method.

This article is in some sense highly technical and the reviewer is not able to check the correctness of all the theorems. Nevertheless reading this article is a much rewarding experience.

Reviewer: B.Richter

##### Keywords:

cubic diophantine equation; computational number theory; reduction algorithm; “small” solutions; continued fraction algorithm; “Very small” solutions; totally real quartic Thue equations
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\textit{N. Tzanakis} and \textit{B. M. M. de Weger}, J. Number Theory 31, No. 2, 99--132 (1989; Zbl 0657.10014)

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