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Catalan's equation $ x\sp p-y\sp q=1$ and related congruences

Authors: M. Aaltonen and K. Inkeri
Journal: Math. Comp. 56 (1991), 359-370
MSC: Primary 11D41; Secondary 11A07, 11D61
MathSciNet review: 1052082
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Abstract: We investigate solutions of Catalan's equation $ {x^p} - {y^q} = 1$ in nonzero integers x, y, p, q. By use of class numbers and congruences $ {p^q} \equiv p\;\pmod {q^2}$ we show the impossibility of the equation for a large number of prime exponents p, q. Both theoretical and computer results are included. We also study lower bounds on possible, hitherto unknown, solutions $ x,y > 2$; we especially wish to communicate the bound $ x,y \geq {10^{500}}$.

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Article copyright: © Copyright 1991 American Mathematical Society