Geodesic
Perfect powers with few binary digits and related Diophantine problems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 941-953

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We prove that, for any fixed base x≥2 and sufficiently large prime q, no perfect q-th power can be written with 3 or 4 digits 1 in base x. This is a particular instance of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms.

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Classification : 11A63, 11D61, 11J86 
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     author = {Bennett, Michael A. and Bugeaud, Yann and Mignotte, Maurice},
     title = {Perfect powers with few binary digits and related {Diophantine} problems},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {941--953},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {4},
     year = {2013},
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     zbl = {1303.11084},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2013_5_12_4_941_0/}
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Bennett, Michael A.; Bugeaud, Yann; Mignotte, Maurice. Perfect powers with few binary digits and related Diophantine problems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 941-953. http://geodesic.mathdoc.fr/item/ASNSP_2013_5_12_4_941_0/