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A class number free criterion for Catalan’s conjecture. (English) Zbl 1049.11036
Catalan’s conjecture [E. Catalan, J. Reine Angew. Math. 27, 192 (1844; ERAM 027.0790cj)] predicts that 8 and 9 are the only consecutive integers which are both perfect powers. This conjecture was recently proved by the present author. This paper contains the first part of his contribution to the final solution of this problem.
Catalan’s conjecture corresponds to the Diophantine equation $$x^p-y^q=1$$ where $$p$$ and $$q$$ are prime numbers. Several arithmetical criteria were obtained by K. Inkeri [J. Number Theory 34, 142–152 (1990; Zbl 0699.10029)] and after M. Mignotte [C. R. Math. Acad. Sci., Soc. R. Can. 15, 199–200 (1993; Zbl 0802.11010)] and W. Schwarz [Acta Arith. 72, 277–279 (1995; Zbl 0837.11014)], but all of them implied some condition on certain class numbers. In this paper the author is able to get rid of these conditions and proves the remarkable fact that if the above equation has a nontrivial solution in rational integers then
$p^{q-1} \equiv 1 \pmod {q^2} \quad \text{ and} \quad q^{p-1} \equiv 1 \pmod {p^2}.$
The proof follows Inkeri’s proof, except for a very ingenious use of Stickelberger’s theorem.

##### MSC:
 11D61 Exponential Diophantine equations
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##### References:
 [1] Baker, A., Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge philos. soc., 65, 439-444, (1969) · Zbl 0174.33803 [2] Baker, A.; Baker, A., A sharpening of the bounds for linear forms in logarithms II, Acta arith., Acta arith., 24, 33-36, (1973) · Zbl 0261.10025 [3] Bennett, C.; Blass, J.; Glass, A.M.W.; Meronk, D.; Steiner, R.P., Linear forms in the logarithms of three positive rational integers, J. théorie nombres Bordeaux, 9, 97-136, (1997) · Zbl 0905.11032 [4] Billu, Y.; Hanrot, G., Solving superelliptic Diophantine equations by Baker’s method, Compositio math., 112, 273-312, (1998) · Zbl 0915.11065 [5] J. Blass, A.M.W. Glass, W. O’Neil, Catalan’s conjecture and linear forms in logarithms, preprint. [6] Y. Bugeaud, G. Hanrot, Un nouveau critère pour l’équation de Catalan, preprint, 1999. [7] J.W.S. Cassels, On the equation ax−by=1, II Proc. Cambridge So. (1960) 97-103. [8] Catalan, E., Quelques theéorèmes empiriques. (Mélanges mathématiques, XV), Mém. soc. royale sci. liège, 12, 42-43, (1885) [9] R. Ernvall, Metsänkylä, On the p-divisibility of Fermat quotients, Math. Comput. 66 (1997) 1353-1365. · Zbl 0903.11002 [10] Glass, A.M.W.; Meronk, D.B.; Okada, T.; Steiner, R., A small contribution to Catalan’s equation, J. number theory, 47, 131-137, (1994) · Zbl 0796.11013 [11] Hyyrö, S., Über die gleichung axm−byn=c und das catalansche problem, Ann. acad. sci. fennicae ser. AI, 355, 50pp, (1964) [12] Inkeri, K., On Catalan’s problem, Acta arith., 9, 285-290, (1964) · Zbl 0127.27102 [13] Inkeri, K., On Catalan’s conjecture, J. number theory, 34, 142-152, (1990) · Zbl 0699.10029 [14] Chao, Ko, On the Diophantine equation x2=yn+1,xy≠ 0, Sci. sinica, 14, 457-460, (1965) · Zbl 0163.04004 [15] M. Langevin, Quelques applications de nouveaux résultats de van der Poorten, Sém Delange Pisot Poitou, Paris, Exp. 4. [16] Lebesgue, V.A., Sur l’impossibilité en nombres entiers de l’équation xm=y2+1, Nouv. ann. math., 9, 178-181, (1850) [17] Mignotte, M., Sur l’équation de Catalan, C.R. acad.sci. Paris ser I, 314, 165-168, (1992) · Zbl 0749.11025 [18] Mignotte, M., A criterion on Catalan’s equation, J. number theory, 52, 280-283, (1995) · Zbl 0829.11016 [19] M. Mignotte, Unpublished, cited by [BH]. [20] M. Mignotte, Catalan’s equation just before 2000, in: M. Jutila, (Ed.), Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland, May 31-June 4, de Gruyter, Berlin, 1999, pp. 247-254. · Zbl 1065.11019 [21] Mignotte, M.; Roy, Y., Catalan’s equation has no new solution with either exponent less than 10651, Exp. math., 4, 259-268, (1995) · Zbl 0857.11012 [22] Mignotte, M.; Reoy, Y., Minorations pour l’èquation de Catalan, C.R.acad.sci. Paris, 324, 377-380, (1997) · Zbl 0887.11018 [23] T. O’Neil, Improved upper bounds on the exponents in Catalan’s equation, Manuscript, 1995. [24] Ribenboim, P., Catalan’s conjecture, (1994), Academic Press New York · Zbl 0866.11027 [25] Schwarz, W., A note on Catalan’s equation, Acta arith., 72, 277-279, (1995) · Zbl 0837.11014 [26] Steiner, R., Class number bounds on Catalan’s equation, Math. comput., 67, 1317-1322, (1998) · Zbl 0897.11009 [27] Tijdeman, R., On the equation of Catalan, Acta arith., 29, 197-209, (1976) [28] L. Washington, Introduction to Cyclotomic Fields, 2nd Edition, Graduate Texts in Mathematics, Vol. 83, Springer, Berlin, 1996. · Zbl 0966.11047
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