Primary cyclotomic units and a proof of Catalan’s conjecture.

*(English)*Zbl 1067.11017Catalan’s conjecture (1844) predicts that 8 and 9 are the only consecutive integers which are both perfect powers. This famous conjecture is proved in this paper. Catalan’s conjecture corresponds to the Diophantine equation \(x^p-y^q=1\), where \(p\) and \(q\) are prime numbers and \(x\), \(y\) are rational integers with \(xy\ne 0\).

The case of \(q=2\) was settled in 1850 by V. A. Lebesgue, there is no solution. But the case \(p=2\) was solved only in 1965, by Ko Chao: the only solution corresponds to \(9-8=1\). Thus, from now on we assume that \(p\) and \(q\) are both odd.

In 1960, using arguments from Diophantine approximation, Cassels proved that, if there is a solution, then \(p\) divides \(y\) and \(q\) divides \(x\). This result easily implies very useful formulas for \(x\) and \(y\), namely

\[ x-1=p^{q-1} a^q, \quad {xp-1 \over x-1}=p^{q-1} u^q \]

for some rational integers \(a\) and \(u\), and similar formulas for \(y\).

Combined with Baker’s theory on linear forms of logarithms, these formulas played an essential rôle in the proof by Tijdeman, in 1976, that Catalan’s problem is a finite problem: the exponents \(p\) and \(q\) are bounded and the same holds for the variables \(x\) and \(y\). But the bounds were far too enormous to lead to a solution assisted by computers.

Several arithmetical criteria were obtained by Inkeri and others between 1964 and 1998. In a previous paper published in 2003 [see J. Number Theory 99, No. 2, 225–231 (2003; Zbl 1049.11036)], the author was able to get rid of the technical conditions appearing in these results, and he proved that if the above equation has a non trivial solution in rational integers then \[ p^{q-1} \equiv 1 \pmod {q^2} \quad \text{ and} \quad q^{p-1} \equiv 1 \pmod {p^2}. \]

Assume now, without loss of generality, that \(p>q\). In the case \(p\equiv 1 \pmod q\), the author could show that there is no solution using his above criterion and inequalities obtained by Mignotte and Roy (1997) using sharp lower bounds for linear forms with two logarithms. Indeed, in this case, one gets \(q<100,000\) and a few minutes of computation, using the previous criterion, show that there is no solution.

In this paper, this is the only place where Baker’s theory is used. Note that in a more recent work, Mihăilescu was able to get rid of this theory and to give a completely “elementary” proof of Catalan’s conjecture. In the remaining case, \(p\not\equiv 1 \pmod q\), a deep theorem of Thaine on cyclotomic fields applies. Thanks to this theorem and to a very deep and subtle study of certain \(G\)-modules, where \(G\) is the Galois group of the cyclotomic field of \(q\)-th roots of unity, Mihăilescu is able (the proof is about 20 pages long) to conclude that there is no solution. I am not able to explain the argument in a short place, but the reader may also look at the presentation given by Y. Bilu in the Bourbaki Seminar of November 2002 [Bourbaki seminar. Volume 2002/2003. Exp. No. 909, Astérisque 294, 1–26 (2004; Zbl 1094.11014)].

The case of \(q=2\) was settled in 1850 by V. A. Lebesgue, there is no solution. But the case \(p=2\) was solved only in 1965, by Ko Chao: the only solution corresponds to \(9-8=1\). Thus, from now on we assume that \(p\) and \(q\) are both odd.

In 1960, using arguments from Diophantine approximation, Cassels proved that, if there is a solution, then \(p\) divides \(y\) and \(q\) divides \(x\). This result easily implies very useful formulas for \(x\) and \(y\), namely

\[ x-1=p^{q-1} a^q, \quad {xp-1 \over x-1}=p^{q-1} u^q \]

for some rational integers \(a\) and \(u\), and similar formulas for \(y\).

Combined with Baker’s theory on linear forms of logarithms, these formulas played an essential rôle in the proof by Tijdeman, in 1976, that Catalan’s problem is a finite problem: the exponents \(p\) and \(q\) are bounded and the same holds for the variables \(x\) and \(y\). But the bounds were far too enormous to lead to a solution assisted by computers.

Several arithmetical criteria were obtained by Inkeri and others between 1964 and 1998. In a previous paper published in 2003 [see J. Number Theory 99, No. 2, 225–231 (2003; Zbl 1049.11036)], the author was able to get rid of the technical conditions appearing in these results, and he proved that if the above equation has a non trivial solution in rational integers then \[ p^{q-1} \equiv 1 \pmod {q^2} \quad \text{ and} \quad q^{p-1} \equiv 1 \pmod {p^2}. \]

Assume now, without loss of generality, that \(p>q\). In the case \(p\equiv 1 \pmod q\), the author could show that there is no solution using his above criterion and inequalities obtained by Mignotte and Roy (1997) using sharp lower bounds for linear forms with two logarithms. Indeed, in this case, one gets \(q<100,000\) and a few minutes of computation, using the previous criterion, show that there is no solution.

In this paper, this is the only place where Baker’s theory is used. Note that in a more recent work, Mihăilescu was able to get rid of this theory and to give a completely “elementary” proof of Catalan’s conjecture. In the remaining case, \(p\not\equiv 1 \pmod q\), a deep theorem of Thaine on cyclotomic fields applies. Thanks to this theorem and to a very deep and subtle study of certain \(G\)-modules, where \(G\) is the Galois group of the cyclotomic field of \(q\)-th roots of unity, Mihăilescu is able (the proof is about 20 pages long) to conclude that there is no solution. I am not able to explain the argument in a short place, but the reader may also look at the presentation given by Y. Bilu in the Bourbaki Seminar of November 2002 [Bourbaki seminar. Volume 2002/2003. Exp. No. 909, Astérisque 294, 1–26 (2004; Zbl 1094.11014)].

Reviewer: Maurice Mignotte (Strasbourg)

##### MSC:

11D61 | Exponential Diophantine equations |

11R18 | Cyclotomic extensions |

11R27 | Units and factorization |

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

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##### References:

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