Catalan's Conjecture
Séminaire de Philosophie et Mathématiques, Catalan's conjecture, no. 6 (1994), pp. 1-11. 
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     author = {Ribenboim, Paulo},
     title = {Catalan's {Conjecture}},
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     publisher = {\'Ecole Normale Sup\'erieure},
     number = {6},
     year = {1994},
     language = {en},
     url = {http://www.numdam.org/item/SPHM_1994___6_A1_0/}
}
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Ribenboim, Paulo. Catalan's Conjecture. Séminaire de Philosophie et Mathématiques, Catalan's conjecture, no. 6 (1994), pp. 1-11. http://www.numdam.org/item/SPHM_1994___6_A1_0/

For proofs, remarks, details concerning Catalan's conjecture, you may consult my own book, which contains a fairly complete bibliography about the problem : P. Ribenboim "Catalan's Conjecture" Academic Press, Boston, 1994.

Previously, I have published a survey on the problem : P. Ribenboim "Consecutive power" Expositiones Mathematicae, 2 (1984), 193-221. | Zbl

There are also several very recent preprints by A.W. Glass et al., by M. Mignotte and by W. Schwarz, which are concerned with developments in the line of K. Inkeri's criteria and the related calculations, some of whch still unpublished.

The results about prime numbers may be found for example in my own book : P. Ribenboim "The Book of Prime Number Records" Springer-Verlag, New York (first editions 1987 ; second edition 1989, third edition 1995).

See also the French abridged version : P. Ribenboim "Les Nombres Premiers : Mystères et Records", Presses Universitaires de France, Paris, 1994. | Zbl