Exponential Diophantine equations.

*(English)*Zbl 0606.10011
Cambridge Tracts in Mathematics, 87. Cambridge etc.: Cambridge University Press. x, 240 p. £25.00; $ 44.50 (1986).

What is an exponential diophantine equation (EDE) ? It is a diophantine equation in which at least one exponent in the equation is unknown. Therefore \(x^2+k=y^3\) is not an EDE, but (*) \(x^2+k=y^n\) is one. Also (**) \(x^x y^y=z^z\) is an EDE.

Perhaps the most famous EDE is the Fermat equation \(x^n+y^n=z^n\). Another well known one is Catalan’s \(x^n-y^m=1\) or the related Pillai’s \(ax^n+by^m=k\) which includes (*) and Catalan’s equation. If you take rational solutions into account (not only integral ones) there also seems to be a connection between Fermat’s and Pillai’s equations; but the first one defines a smooth projective curve and the second one has a singularity at infinity for \(n\neq m\). So, one way to get results is the algebro-geometric way with his recent achievements by Faltings’ theorem. But there always the exponents must be fixed.

The authors’ aim is to give explicitly computable upper bounds for the unknown exponents in the EDE’s so that one should be able to apply the known results for polynomial diophantine equations. But in almost all cases examined these upper bounds are astronomical large and not reachable even by the fastest computers. – I think all people involved with these questions believe that there must be small upper bounds, if there are any, but this seems beyond the powers of the present methods. – So the authors introduce the concept of computable number which doesn’t give hope that really all solutions can be written down: A computable number is a positive real number which you can effectively calculate following the way of the given proof (it may depend on the parameters of the problem).

The authors refuse to compute any of these ”computable number”. Even for the constant in the second named author’s famous proof that for all positive integers \(x,y,n,m\) with \(x^n-y^m=1\) it follows that \(x,y,n,m<C_1\) an explicit value for \(C_1\) is not given. Only in the notes you can find that \(x^n<\exp\exp\exp\exp (730)\) and that the largest prime factor of \(mn\) is smaller than \(\exp(241)\) [M. Langevin, Sémin. Delange-Pisot-Poitou, 17e Année 1975/76, Paris, Fasc. 2, Exp. No. G 12 (1977; Zbl 0354.10008)].

Nearly all methods for solving the mentioned equations also apply to (integers in) algebraic number fields. The corresponding results for algebraic function fields are mostly deported to the notes.

The main method of proof is Baker’s method of linear forms in logarithms of algebraic numbers [chiefly A. Baker, Mathematika 15, 204-216 (1968; Zbl 0169.37802)]. Recently the first author also applied Baker’s rational approximation results to algebraic numbers [A. Baker, Q. J. Math., Oxf. II. Ser. 15, 375–383 (1964; Zbl 0222.10036)] in a paper concerned with Pillai’s equation [Indagationes Math. 48, 353–358 (1986; Zbl 0603.10019)]. The book opens after a lucid introduction with three preliminary chapters which contain no proof.

Chapter A: Results from algebraic number theory. It gives a review of algebraic number theory with special emphasis on computable results, e.g. E. Landau [Gött. Nachr. 1918, 478–488 (1918; JFM 46.0267.01)]: Class number \(h\) and regulator \(R\) of an algebraic number field \(K\) of degree \(d\) over \({\mathbb{Q}}\) are bounded by a computable constant \(C_2\), dependent only on \(d\), such that \[ hR < C_2 \sqrt{| D|} (\log | D|)^{d-1} \] where \(D\) is the discriminant of \(K\) over \({\mathbb{Q}}\), or R. Zimmert [Invent. Math. 62, 367–380 (1980; Zbl 0456.12003)]: \(R>0.056\) (one of the few explicit constants in this monograph).

Chapter B: Estimates of linear forms in logarithms (3 pages). It is devoted to Baker’s method and the related results of van der Poorten in the \(p\)-adic case. It seems a pity for the reviewer that for these far-reaching methods which are central not only for EDE’s no proofs are given. The reader is referred to the original papers. A reference to S. Lang [Elliptic curves: Diophantine analysis (1978; Zbl 0388.10001)] is missing at this place. (Besides, the best lower bound for linear forms in logarithms of algebraic numbers seems to be in the contribution by J. H. Loxton, M. Mignotte, A. J. van der Poorten and M. Waldschmidt [C. R. Math. Acad. Sci., Soc. R. Can. 11, 119–124 (1987; Zbl 0623.10023)].)

Chapter C: Recurrence sequences. On the first view this chapter seems not to be related with the theory of EDE’s. But as the authors explain a good theory of EDE’s should give, for instance, all perfect powers in the Fibonacci or Lucas series. (This will be done in chapter 9 of the main part of the book.)

We come now to the gist of the book which is organized in 12 chapters. One thing where I believe to realize again the mathematical style of the first author is the set-up of the individual chapters: Each chapter is in three parts. The first part contains the statements of all the results to be proved in the chapter. The second part contains the proofs of these results. The third part gives an account of the developments related to the results of the chapter. Thus an account of the results of important topics which are not included in the text are available in the notes. Because of lack of place we will only rush through the chapters.

Chapter 1 is concerned with purely exponential equations. Here only the exponents are unknown. A special problem is the equation \(x+y=z\) where \(x, y, z\) are integers composed of primes from a given finite set. There are a lot of results which have been obtained by elementary methods, especially by evaluating congruences modulo larger and larger primes. This method is justified by corollary 1.2.: \(\max (|x|, |y|, |z|)\) is bounded by a computable number dependent on the largest prime factor of \(xyz\). (For additional elementary results I refer, e.g., to my reviews of A. Grytczuk and A. Grelak [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 24, 269–275 (1984; Zbl 0571.10017)] where the equation \(a^n+b^m=c^\ell\) is considered \((a, b, c\) fixed integers), or of Yen Chente [Bull. Inst. Math., Acad. Sin. 13, 49–92 (1985; Zbl 0566.10012)].)

For the special exponential diophantine equation \(3^n+13^m=2^\ell\) S. Uchiyama was forced to develop another method depending on recurrence relations in the number field \({\mathbb{Q}}(\sqrt{-39})\) [Math. J. Okayama Univ. 19, 31–38 (1976; Zbl 0349.10013)], a result which is not mentioned by the authors. Another equation which also falls into the authors’ scope is not tackled by them: the equation (**), which is not even mentioned. I again refer to S. Uchiyama [Tr. Mat. Inst. Steklova 163, 237–243 (1984; Zbl 0551.10016)] where he gives a short overview of the present stage of investigation (which also includes applications of Baker’s method) and some new insights.

Chapters 2 to 4 and 9 are concerned with recurrence sequences. I will only quote two characteristic examples:

(i) [the second author, Sémin. Delange-Pisot-Poitou, 16e année 1974/75, Théorie des Nombres, Fasc. 2, Exp. No. 24 (1975; Zbl 0319.10022)]: If \(a, b, k, m, x, y\in {\mathbb{Z}}\), \(a, b, k\neq 0\), \(m, y\geq 0\), \(x>1\) and \[ ax^m+by^m=k \] then \(m\) is bounded by a computable number depending only on \(a, b, k\) (remember Pillai’s equation).

(ii) [M. Ram Murty, V. Kumar Murty and the first author [Bull. Soc. Math. Fr. 115, 391–395 (1987; Zbl 0635.10020)]: Let \(\tau(n)\) be Ramanujan’s function. It satisfies a binary recurrence \[ \tau(p^{m+1})=\tau(p) \tau(p^ m)-p^{11} \tau(p^{m-1})\quad (m=1,2,...). \]

Theorem: For an odd integer \(a\) the equation \(\tau(n)=a\) implies \(\log n\leq (2|a|)^{C_3}\) where \(C_3\) is a computable absolute constant.

The remaining chapters contain the well known results, mostly by the authors and K. Győry, on the Thue equation, the Thue-Mahler equation (we refer to the book for explanation of the difference), the superelliptic equation (as (*)), and on Fermat’s and Catalan’s equations. There is also a chapter on perfect powers at integral values of a polynomial, a central theme which is perhaps the core of the book. Everybody interested in this peculiar (or main ?) field of the theory of diophantine equations will be glad to have these results and proofs at hand and must not dig through the wealth of literature.

I will now give some more additions to the extended bibliography (18 pages with approximately 30 entries per page). This bibliography has the great advantage of giving the chapter where each entry is cited.

In the last chapter, mainly involved with the Catalan-Pillai equation, there are also some applications to the equation \[ \frac{x^ n-1}{x- 1}=y^ m. \] Here, the first author’s paper [Indagationes Math. 48, 345–351 (1986; Zbl 0603.10018)] has appeared. There is a substantial contribution by D. Estes, R. Guralnick, M. Schacher and E. Straus [Pac. J. Math. 118, 359–367 (1985; Zbl 0581.20009)] to this equation in prime powers \(x, y\) with applications to finite simple groups. In the quotation of the reviewer’s paper [Acta Arith. 40, 273–288 (1982; Zbl 0405.10014)] one should change the ”\(a\)” on the right side to ”\(\alpha\) ” (also in the citation on p. 218 of the book – as well as in Zbl 0474.10017).

There are now very interesting papers of É. Fouvry [Invent. Math. 79, 383–407 (1985; Zbl 0557.10035)] and L. M. Adleman and D. R. Heath-Brown [Invent. Math. 79, 409–416 (1985; Zbl 0557.10034)] on the density of primes for which the first case of Fermat’s theorem is valid. Cf. also D. R. Heath-Brown [Math. Intell. 7, No. 4, 40–47, 55 (1985; Zbl 0574.10022)]. Density results for all natural numbers as exponents in Fermat’s theorem have been given by B. Powell and P. Ribenboim [Ann. Univ. Turku, Ser. A I 187, 22 p. (1985; Zbl 0564.10014)].

By these methods which are actually combinatorial (sieve methods) P. Ribenboim obtains in Pillai’s equation [J. Number Theory 24, 245–248 (1986; Zbl 0601.10013)] for all \(a, b, k\in\mathbb{Z}\setminus \{0\}\) and \(N\in \mathbb{N}\) with \[D_N=\#\{(n,m)\in\mathbb{N}^2 \mid 1\leq n, m\leq N:\ \forall x,y\in \mathbb{Z},\ ax^n+by^m\neq k\}\] that \[\lim_{N\to \infty}(D_ N/N^ 2)=1.\]

One should note that the results of Heath-Brown et al. are ultimate contributions to sieve methods whereas Ribenboim’s method is essentially that of Eratosthenes [cf. also P. Ribenboim’s survey on recent results about Fermat’s last theorem, Expo. Math. 5, 75–90 (1987; Zbl 0604.10005)].

A minor misprint is ”\(2Y^3\)” instead of ”\(3Y^3\)” in N. Tzanakis [J. Number Theory 19, 203–208 (1984; Zbl 0543.10016)].

All this is not serious because a good mathematics database such as zbMATH online will overcome these troubles. The only substantial mistake in the reviewer’s opinion seems to be the citation of C. Størmer [C. R. Acad. Sci., Paris 127, 752–754 (1898)] in chapter 1 but not in chapter 12 because he was the first one who showed that for fixed \((x,y)\neq (2,3)\) in Catalan’s equation there is at most one solution \(m, n\) (effectively determined by \(x, y)\) (cf. the review of W. Ljunggren in Zbl 0047.04103 to the paper of W. J. LeVeque Am. J. Math. 74, 325–331 (1952)].) I believe one should read this author anew.

Perhaps the most famous EDE is the Fermat equation \(x^n+y^n=z^n\). Another well known one is Catalan’s \(x^n-y^m=1\) or the related Pillai’s \(ax^n+by^m=k\) which includes (*) and Catalan’s equation. If you take rational solutions into account (not only integral ones) there also seems to be a connection between Fermat’s and Pillai’s equations; but the first one defines a smooth projective curve and the second one has a singularity at infinity for \(n\neq m\). So, one way to get results is the algebro-geometric way with his recent achievements by Faltings’ theorem. But there always the exponents must be fixed.

The authors’ aim is to give explicitly computable upper bounds for the unknown exponents in the EDE’s so that one should be able to apply the known results for polynomial diophantine equations. But in almost all cases examined these upper bounds are astronomical large and not reachable even by the fastest computers. – I think all people involved with these questions believe that there must be small upper bounds, if there are any, but this seems beyond the powers of the present methods. – So the authors introduce the concept of computable number which doesn’t give hope that really all solutions can be written down: A computable number is a positive real number which you can effectively calculate following the way of the given proof (it may depend on the parameters of the problem).

The authors refuse to compute any of these ”computable number”. Even for the constant in the second named author’s famous proof that for all positive integers \(x,y,n,m\) with \(x^n-y^m=1\) it follows that \(x,y,n,m<C_1\) an explicit value for \(C_1\) is not given. Only in the notes you can find that \(x^n<\exp\exp\exp\exp (730)\) and that the largest prime factor of \(mn\) is smaller than \(\exp(241)\) [M. Langevin, Sémin. Delange-Pisot-Poitou, 17e Année 1975/76, Paris, Fasc. 2, Exp. No. G 12 (1977; Zbl 0354.10008)].

Nearly all methods for solving the mentioned equations also apply to (integers in) algebraic number fields. The corresponding results for algebraic function fields are mostly deported to the notes.

The main method of proof is Baker’s method of linear forms in logarithms of algebraic numbers [chiefly A. Baker, Mathematika 15, 204-216 (1968; Zbl 0169.37802)]. Recently the first author also applied Baker’s rational approximation results to algebraic numbers [A. Baker, Q. J. Math., Oxf. II. Ser. 15, 375–383 (1964; Zbl 0222.10036)] in a paper concerned with Pillai’s equation [Indagationes Math. 48, 353–358 (1986; Zbl 0603.10019)]. The book opens after a lucid introduction with three preliminary chapters which contain no proof.

Chapter A: Results from algebraic number theory. It gives a review of algebraic number theory with special emphasis on computable results, e.g. E. Landau [Gött. Nachr. 1918, 478–488 (1918; JFM 46.0267.01)]: Class number \(h\) and regulator \(R\) of an algebraic number field \(K\) of degree \(d\) over \({\mathbb{Q}}\) are bounded by a computable constant \(C_2\), dependent only on \(d\), such that \[ hR < C_2 \sqrt{| D|} (\log | D|)^{d-1} \] where \(D\) is the discriminant of \(K\) over \({\mathbb{Q}}\), or R. Zimmert [Invent. Math. 62, 367–380 (1980; Zbl 0456.12003)]: \(R>0.056\) (one of the few explicit constants in this monograph).

Chapter B: Estimates of linear forms in logarithms (3 pages). It is devoted to Baker’s method and the related results of van der Poorten in the \(p\)-adic case. It seems a pity for the reviewer that for these far-reaching methods which are central not only for EDE’s no proofs are given. The reader is referred to the original papers. A reference to S. Lang [Elliptic curves: Diophantine analysis (1978; Zbl 0388.10001)] is missing at this place. (Besides, the best lower bound for linear forms in logarithms of algebraic numbers seems to be in the contribution by J. H. Loxton, M. Mignotte, A. J. van der Poorten and M. Waldschmidt [C. R. Math. Acad. Sci., Soc. R. Can. 11, 119–124 (1987; Zbl 0623.10023)].)

Chapter C: Recurrence sequences. On the first view this chapter seems not to be related with the theory of EDE’s. But as the authors explain a good theory of EDE’s should give, for instance, all perfect powers in the Fibonacci or Lucas series. (This will be done in chapter 9 of the main part of the book.)

We come now to the gist of the book which is organized in 12 chapters. One thing where I believe to realize again the mathematical style of the first author is the set-up of the individual chapters: Each chapter is in three parts. The first part contains the statements of all the results to be proved in the chapter. The second part contains the proofs of these results. The third part gives an account of the developments related to the results of the chapter. Thus an account of the results of important topics which are not included in the text are available in the notes. Because of lack of place we will only rush through the chapters.

Chapter 1 is concerned with purely exponential equations. Here only the exponents are unknown. A special problem is the equation \(x+y=z\) where \(x, y, z\) are integers composed of primes from a given finite set. There are a lot of results which have been obtained by elementary methods, especially by evaluating congruences modulo larger and larger primes. This method is justified by corollary 1.2.: \(\max (|x|, |y|, |z|)\) is bounded by a computable number dependent on the largest prime factor of \(xyz\). (For additional elementary results I refer, e.g., to my reviews of A. Grytczuk and A. Grelak [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 24, 269–275 (1984; Zbl 0571.10017)] where the equation \(a^n+b^m=c^\ell\) is considered \((a, b, c\) fixed integers), or of Yen Chente [Bull. Inst. Math., Acad. Sin. 13, 49–92 (1985; Zbl 0566.10012)].)

For the special exponential diophantine equation \(3^n+13^m=2^\ell\) S. Uchiyama was forced to develop another method depending on recurrence relations in the number field \({\mathbb{Q}}(\sqrt{-39})\) [Math. J. Okayama Univ. 19, 31–38 (1976; Zbl 0349.10013)], a result which is not mentioned by the authors. Another equation which also falls into the authors’ scope is not tackled by them: the equation (**), which is not even mentioned. I again refer to S. Uchiyama [Tr. Mat. Inst. Steklova 163, 237–243 (1984; Zbl 0551.10016)] where he gives a short overview of the present stage of investigation (which also includes applications of Baker’s method) and some new insights.

Chapters 2 to 4 and 9 are concerned with recurrence sequences. I will only quote two characteristic examples:

(i) [the second author, Sémin. Delange-Pisot-Poitou, 16e année 1974/75, Théorie des Nombres, Fasc. 2, Exp. No. 24 (1975; Zbl 0319.10022)]: If \(a, b, k, m, x, y\in {\mathbb{Z}}\), \(a, b, k\neq 0\), \(m, y\geq 0\), \(x>1\) and \[ ax^m+by^m=k \] then \(m\) is bounded by a computable number depending only on \(a, b, k\) (remember Pillai’s equation).

(ii) [M. Ram Murty, V. Kumar Murty and the first author [Bull. Soc. Math. Fr. 115, 391–395 (1987; Zbl 0635.10020)]: Let \(\tau(n)\) be Ramanujan’s function. It satisfies a binary recurrence \[ \tau(p^{m+1})=\tau(p) \tau(p^ m)-p^{11} \tau(p^{m-1})\quad (m=1,2,...). \]

Theorem: For an odd integer \(a\) the equation \(\tau(n)=a\) implies \(\log n\leq (2|a|)^{C_3}\) where \(C_3\) is a computable absolute constant.

The remaining chapters contain the well known results, mostly by the authors and K. Győry, on the Thue equation, the Thue-Mahler equation (we refer to the book for explanation of the difference), the superelliptic equation (as (*)), and on Fermat’s and Catalan’s equations. There is also a chapter on perfect powers at integral values of a polynomial, a central theme which is perhaps the core of the book. Everybody interested in this peculiar (or main ?) field of the theory of diophantine equations will be glad to have these results and proofs at hand and must not dig through the wealth of literature.

I will now give some more additions to the extended bibliography (18 pages with approximately 30 entries per page). This bibliography has the great advantage of giving the chapter where each entry is cited.

In the last chapter, mainly involved with the Catalan-Pillai equation, there are also some applications to the equation \[ \frac{x^ n-1}{x- 1}=y^ m. \] Here, the first author’s paper [Indagationes Math. 48, 345–351 (1986; Zbl 0603.10018)] has appeared. There is a substantial contribution by D. Estes, R. Guralnick, M. Schacher and E. Straus [Pac. J. Math. 118, 359–367 (1985; Zbl 0581.20009)] to this equation in prime powers \(x, y\) with applications to finite simple groups. In the quotation of the reviewer’s paper [Acta Arith. 40, 273–288 (1982; Zbl 0405.10014)] one should change the ”\(a\)” on the right side to ”\(\alpha\) ” (also in the citation on p. 218 of the book – as well as in Zbl 0474.10017).

There are now very interesting papers of É. Fouvry [Invent. Math. 79, 383–407 (1985; Zbl 0557.10035)] and L. M. Adleman and D. R. Heath-Brown [Invent. Math. 79, 409–416 (1985; Zbl 0557.10034)] on the density of primes for which the first case of Fermat’s theorem is valid. Cf. also D. R. Heath-Brown [Math. Intell. 7, No. 4, 40–47, 55 (1985; Zbl 0574.10022)]. Density results for all natural numbers as exponents in Fermat’s theorem have been given by B. Powell and P. Ribenboim [Ann. Univ. Turku, Ser. A I 187, 22 p. (1985; Zbl 0564.10014)].

By these methods which are actually combinatorial (sieve methods) P. Ribenboim obtains in Pillai’s equation [J. Number Theory 24, 245–248 (1986; Zbl 0601.10013)] for all \(a, b, k\in\mathbb{Z}\setminus \{0\}\) and \(N\in \mathbb{N}\) with \[D_N=\#\{(n,m)\in\mathbb{N}^2 \mid 1\leq n, m\leq N:\ \forall x,y\in \mathbb{Z},\ ax^n+by^m\neq k\}\] that \[\lim_{N\to \infty}(D_ N/N^ 2)=1.\]

One should note that the results of Heath-Brown et al. are ultimate contributions to sieve methods whereas Ribenboim’s method is essentially that of Eratosthenes [cf. also P. Ribenboim’s survey on recent results about Fermat’s last theorem, Expo. Math. 5, 75–90 (1987; Zbl 0604.10005)].

A minor misprint is ”\(2Y^3\)” instead of ”\(3Y^3\)” in N. Tzanakis [J. Number Theory 19, 203–208 (1984; Zbl 0543.10016)].

All this is not serious because a good mathematics database such as zbMATH online will overcome these troubles. The only substantial mistake in the reviewer’s opinion seems to be the citation of C. Størmer [C. R. Acad. Sci., Paris 127, 752–754 (1898)] in chapter 1 but not in chapter 12 because he was the first one who showed that for fixed \((x,y)\neq (2,3)\) in Catalan’s equation there is at most one solution \(m, n\) (effectively determined by \(x, y)\) (cf. the review of W. Ljunggren in Zbl 0047.04103 to the paper of W. J. LeVeque Am. J. Math. 74, 325–331 (1952)].) I believe one should read this author anew.

Reviewer: Bernd Richter (Berlin)

##### MSC:

11D61 | Exponential Diophantine equations |

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

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

11D88 | \(p\)-adic and power series fields |

11R58 | Arithmetic theory of algebraic function fields |