Perfect powers with few binary digits and related Diophantine problems
Michael A. Bennett, Yann Bugeaud, Maurice Mignotte · ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE · 2013
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. Mathematics Subject Classification (2010): 11A63 (primary); 11D61, 11J86 (secondary).commonly termed the Nagell-Ljunggren equation, has no solutions beyond the three listed above.Remarkably, with the current technology, it is not even known whether this equation has finitely many solutions in the four variables.For a given fixed value of x, however, it is always possible to solve (1.1), at least in principle;