Algorithms for diophantine equations.

*(English)*Zbl 0687.10013
CWI Tract, 65. Amsterdam: Centrum voor Wiskunde en Informatica. viii, 212 p. Dfl. 33.00 (1989).

The book is an updated version of the author’s excellent Ph.D. thesis written in 1987. The purpose of it is to give a systematic, computer oriented overview of certain algorithms, applicable for the complete resolution of several types of exponential and polynomial diophantine equations. The algorithms are based on the combination of Baker’s method with a numerical reduction procedure discovered originally by A. Baker and H. Davenport [Q. J. Math., Oxf. II. Ser. 20, 129–137 (1969; Zbl 0177.06802)]. In the book several new and useful variants of this method are given. The algorithms presented are illustrated with interesting numerical examples. At the end of the book an update list of references is given on the theory and practical solution of the equations involved.

Chapter 1 is an introductory explanation of Baker’s method, its \(p\)-adic analogue and the diophantine approximation techniques, frequently used in the book.

Chapter 2 gives the necessary basics from algebraic number theory and the estimates of Waldschmidt, Schinzel and Yu for linear forms in logarithms of algebraic numbers.

Chapter 3 is a detailed description of the computational methods for the reduction of the large upper bounds obtained by Baker’s method for the solutions of diophantine equations. This chapter explains among others the use of the continued fraction algorithm, the lattice basis reduction algorithm of A. Lenstra, H. Lenstra and L. Lovász [Math. Ann. 261, 515–534 (1982; Zbl 0488.12001)], the Fincke-Pohst algorithm for finding short lattice vectors [U. Fincke and M. Pohst, Math. Comput. 44, 463–471 (1985; Zbl 0556.10022)], and the Davenport lemma with its generalizations.

Chapter 4 is devoted to the equation \[ G_n=wp_1^{m_1}\cdots p_s^{m_s}\quad\text{ in } n\in\mathbb{N},\quad 0\leq m_1,\ldots,m_s\in \mathbb{Z} \] where \(G_n\) is a second order linear recurrence sequence, \(w\) is a given positive integer and \(p_1,\ldots,p_s\) are prime numbers. As an application, the generalized Ramanujan-Nagell equation \[ x^2+k=p_1^{z_1}\cdots p_s^{z_s}\quad\text{ in } x\in\mathbb{N},\quad 0\leq z_1,\ldots,z_s\in\mathbb{Z} \] is studied (with a fixed integer \(k)\).

In the following let \(S\) denote the set of integers composed of a finite number of prime numbers.

In Chapter 5 the equation \[ 0<x-y<y^{\delta}\quad\text{ in } x,y\in S \] is solved \((0<\delta <1\) is fixed). The numerical example given in this chapter extends a result of R. J. Stroeker and R. Tijdeman [Computational methods in Number Theory 2, Math. Cent. Tracts 155, 321–369 (1982; Zbl 0521.10013)].

In Chapter 6 \(S\)-unit equations of the type \(x+y=z\), \(x,y,z\in S\) are discussed. These equations are essential for several types of exponential and polynomial diophantine equations. The results of Chapter 5 are used here. The numerical example stated in the chapter gives interesting information about the Oesterle-Masser conjecture (abc conjecture).

In Chapter 7 the resolution of the equation \(x+y=z^2\), \(x,y\in S\), \(z\in\mathbb{N}\), \((x,y)\) square-free is explained, which has applications in arithmetic algebraic geometry.

Finally, in Chapter 8 the Thue equations \(F(x,y)=m\) in \(x,y\in\mathbb{Z}\) are examined, where \(F\) is an irreducible binary form of degree \(\geq 3\) and \(m\) is a non-zero integer. Two quartic Thue equations are solved completely, which are applied to find all integer points on an elliptic curve.

Chapter 1 is an introductory explanation of Baker’s method, its \(p\)-adic analogue and the diophantine approximation techniques, frequently used in the book.

Chapter 2 gives the necessary basics from algebraic number theory and the estimates of Waldschmidt, Schinzel and Yu for linear forms in logarithms of algebraic numbers.

Chapter 3 is a detailed description of the computational methods for the reduction of the large upper bounds obtained by Baker’s method for the solutions of diophantine equations. This chapter explains among others the use of the continued fraction algorithm, the lattice basis reduction algorithm of A. Lenstra, H. Lenstra and L. Lovász [Math. Ann. 261, 515–534 (1982; Zbl 0488.12001)], the Fincke-Pohst algorithm for finding short lattice vectors [U. Fincke and M. Pohst, Math. Comput. 44, 463–471 (1985; Zbl 0556.10022)], and the Davenport lemma with its generalizations.

Chapter 4 is devoted to the equation \[ G_n=wp_1^{m_1}\cdots p_s^{m_s}\quad\text{ in } n\in\mathbb{N},\quad 0\leq m_1,\ldots,m_s\in \mathbb{Z} \] where \(G_n\) is a second order linear recurrence sequence, \(w\) is a given positive integer and \(p_1,\ldots,p_s\) are prime numbers. As an application, the generalized Ramanujan-Nagell equation \[ x^2+k=p_1^{z_1}\cdots p_s^{z_s}\quad\text{ in } x\in\mathbb{N},\quad 0\leq z_1,\ldots,z_s\in\mathbb{Z} \] is studied (with a fixed integer \(k)\).

In the following let \(S\) denote the set of integers composed of a finite number of prime numbers.

In Chapter 5 the equation \[ 0<x-y<y^{\delta}\quad\text{ in } x,y\in S \] is solved \((0<\delta <1\) is fixed). The numerical example given in this chapter extends a result of R. J. Stroeker and R. Tijdeman [Computational methods in Number Theory 2, Math. Cent. Tracts 155, 321–369 (1982; Zbl 0521.10013)].

In Chapter 6 \(S\)-unit equations of the type \(x+y=z\), \(x,y,z\in S\) are discussed. These equations are essential for several types of exponential and polynomial diophantine equations. The results of Chapter 5 are used here. The numerical example stated in the chapter gives interesting information about the Oesterle-Masser conjecture (abc conjecture).

In Chapter 7 the resolution of the equation \(x+y=z^2\), \(x,y\in S\), \(z\in\mathbb{N}\), \((x,y)\) square-free is explained, which has applications in arithmetic algebraic geometry.

Finally, in Chapter 8 the Thue equations \(F(x,y)=m\) in \(x,y\in\mathbb{Z}\) are examined, where \(F\) is an irreducible binary form of degree \(\geq 3\) and \(m\) is a non-zero integer. Two quartic Thue equations are solved completely, which are applied to find all integer points on an elliptic curve.

Reviewer: István Gaál (Debrecen)

##### MSC:

11Y50 | Computer solution of Diophantine equations |

11D61 | Exponential Diophantine equations |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11D41 | Higher degree equations; Fermat’s equation |

11D59 | Thue-Mahler equations |

11J86 | Linear forms in logarithms; Baker’s method |