Inkeri, K. On Catalan’s conjecture. (English) Zbl 0699.10029 J. Number Theory 34, No. 2, 142-152 (1990). The paper contains some results on Catalan’s equation (*) \(x^ p-y^ q=1\) in nonzero integers x, y and odd primes p, q. Let \(h_ m\) be the class number of the cyclotomic field \(K_ m={\mathbb{Q}}(e^{2\pi i/m}).\) Theorem: Let (x,y,p,q) be a solution of (*). If \(q\nmid h_ p\), then \(x\equiv 0 (mod q^ 2)\) and \(p^ q\equiv p (mod q^ 2)\) and if \(p\nmid h_ q\), then \(y\equiv 0 (mod p^ 2)\) and \(q^ p\equiv q (mod p^ 2)\). In his proof, the author uses Cassels’ result q \(| x\), p \(| y\) and some arithmetic in the fields \(K_ p\), \(K_ q\). The author gives several applications of the theorem stated above and previous results [e.g., the author, Acta Arith. 9, 285-290 (1964; Zbl 0127.271)]. For instance, he proves that (*) has no solutions with \(p,q<89\). Reviewer: J.-H.Evertse Cited in 4 ReviewsCited in 7 Documents MSC: 11D61 Exponential Diophantine equations Keywords:Catalan’s equation Citations:Zbl 0127.271 PDFBibTeX XMLCite \textit{K. Inkeri}, J. Number Theory 34, No. 2, 142--152 (1990; Zbl 0699.10029) Full Text: DOI References: [1] Borevich, Z. I.; Shafarevich, J. R., (Number Theory (1966), Academic Press: Academic Press New York/London) · Zbl 0145.04902 [2] Brillhart, J.; Tonascia, J.; Weinberger, P., On the Fermat quotient, (Atkin, A. O.L; Birch, B. J., Computers in Number Theory (1971), Academic Press: Academic Press New York/London) · Zbl 0217.03203 [3] Cassels, J. M.S, On the equation \(a^x\) − \(b^y = 1\), II, (Proc. Cambridge Philos. Soc., 56 (1960)), 97-103 · Zbl 0094.25702 [4] Hasse, H., (Number Theory (1980), Academic-Verlag: Academic-Verlag New York/Berlin) · JFM 63.0116.02 [5] Hilbert, D., Die Theorie der algebraischen Zahlkörper, (Gesammelte Abhandlungen, Vol. I (1965)) · JFM 28.0157.05 [6] Hyyrö, S., Über das Catalansche Problem, Ann. Univ. Turku Ser. A I, 79 (1964) · Zbl 0127.01904 [7] Inkeri, K., On Catalan’s problem, Acta Arith., 9, 285-290 (1964) · Zbl 0127.27102 [8] van der Linden, F., Class number computations of real abelian number fields (1980), University of Amsterdam, preprint · Zbl 0505.12010 [9] Niewiadomski, R., Zur Fermatschen Vermutung, Prace Mat. Fitzyczne, 42 (1935) · JFM 61.1057.05 [10] \( \textsc{B. Oriat}Qa\); \( \textsc{B. Oriat}Qa\) · Zbl 0391.12003 [11] Ribenboim, P., Consecutive powers, Exposition. Math., 2, 193-221 (1984) · Zbl 0537.10009 [12] Riesel, H., Note on the congruence \(a^{p − 1}\) ≡ 1 (mod \(p^2)\), Math. Comp., 18, 149-150 (1964) · Zbl 0119.03903 [13] Tijdeman, R., On the equation of Catalan, Acta Arith., 29, 197-209 (1976) · Zbl 0286.10013 [14] Washington, L. C., (Introduction to Cyclotomic Fields (1982), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0484.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.