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

We prove that, for any fixed base x2 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.

Published online : 2019-02-21
Classification:  11A63,  11D61,  11J86
@article{ASNSP_2013_5_12_4_941_0,
     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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {4},
     year = {2013},
     pages = {941-953},
     zbl = {1303.11084},
     mrnumber = {3184574},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_941_0}
}
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, Serie 5, Volume 12 (2013) no. 4, pp. 941-953. http://www.numdam.org/item/ASNSP_2013_5_12_4_941_0/

[1] M. A. Bennett, Y. Bugeaud and M. Mignotte, Perfect powers with few binary digits and related Diophantine problems, II, Math. Proc. Cambridge Philos. Soc. 153 (2012), 525–540. | MR 2990629 | Zbl 1291.11016

[2] Y. Bugeaud, Linear forms in p-adic logarithms and the Diophantine equation (x n -1)/ (x-1)=y q , Math. Proc. Cambridge Philos. Soc. 127 (1999), 373–381. | MR 1713116 | Zbl 0940.11019

[3] Y. Bugeaud, Linear forms in two m-adic logarithms and applications to Diophantine problems, Compositio Math. 132 (2002), 137–158. | MR 1915172 | Zbl 1171.11318

[4] Y. Bugeaud, M. Cipu and M. Mignotte, On the representation of Fibonacci and Lucas numbers in an integer base, Ann. Math. Qué. 37 (2013), 31–43. | MR 3117736

[5] Y. Bugeaud and M. Laurent, Minoration effective de la distance p-adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), 311–342. | MR 1423057 | Zbl 0870.11045

[6] Y. Bugeaud and M. Mignotte, Sur l’équation diophantienne x n -1 x-1=y q , II, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 741–744. | MR 1694122 | Zbl 0946.11008

[7] Y. Bugeaud and M. Mignotte, L’équation de Nagell–Ljunggren x n -1 x-1=y q , Enseign. Math. (2) 48 (2002), 147–168. | MR 1923422 | Zbl 1040.11015

[8] Y. Bugeaud and M. Mignotte, On the Diophantine equation x n -1 x-1=y q with negative x, In: “Number theory for the millennium”, I (Urbana, IL, 2000), 145–151, A K Peters, Natick, MA, 2002. | MR 1956223 | Zbl 1031.11019

[9] Y. Bugeaud, M. Mignotte and Y. Roy, On the Diophantine equation (x n -1)/(x-1)=y q , Pacific J. Math. 193 (2000), 257–268. | MR 1755817 | Zbl 1064.11030

[10] P. Corvaja and U. Zannier, On the Diophantine equation f(a m ,y)=b n , Acta Arith. 94 (2000), 25–40. | MR 1762454 | Zbl 0963.11020

[11] P. Corvaja and U. Zannier, Application of the subspace theorem to certain Diophantine problems, In: “Diophantine Approximation”, H. E. Schlickewei et al. (eds.), Springer-Verlag, 2008, 161–174. | MR 2487692 | Zbl 1245.11086

[12] P. Corvaja and U. Zannier, Finiteness of odd perfect powers with four nonzero binary digits, preprint. | Numdam | MR 3112846 | Zbl 1294.11117

[13] M. Hindry and J. Silverman, “Diophantine Geometry, An Introduction”, Springer Verlag GTM 201, 2001. | MR 1745599 | Zbl 0948.11023

[14] J. Lagarias, Ternary expansions of powers of 2, J. London Math. Soc. 79 (2009), 562–588. | MR 2506687 | Zbl 1193.11008

[15] M. Laurent, Linear forms in two logarithms and interpolation determinants. II, Acta Arith. 133 (2008), 325–348. | MR 2457264 | Zbl 1215.11074

[16] F. Luca, The Diophantine equation x 2 =p a ±p b +1, Acta Arith. 112 (2004), 87–101. | MR 2040594 | Zbl 1067.11016

[17] M. Mignotte, Sur les entiers qui s’écrivent simplement en différentes bases, European J. Combin. 9 (1988), 307–316. | MR 950050 | Zbl 0668.10022

[18] P. Mihăilescu, Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine Angew. Math. 572 (2004), 167–195. | MR 2076124 | Zbl 1067.11017

[19] R. Scott, Elementary treatment of p a ±p b +1=x 2 , available online at the homepage of Robert Styer : http://www41.homepage.villanova.edu/robert.styer/ReeseScott/index.htm.

[20] H. G. Senge and E. G. Straus, PV -numbers and sets of multiplicity, Period. Math. Hungar. 3 (1973), 93–100. | MR 340185 | Zbl 0248.12004

[21] C. L. Stewart, On the representation of an integer in two different bases, J. Reine Angew. Math. 319 (1980), 63–72. | MR 586115 | Zbl 0426.10008

[22] L. Szalay, The equations 2 n ±2 m ±2 l =z 2 , Indag. Math. (N.S.) 13 (2002), 131–142. | MR 2014980 | Zbl 1014.11022

[23] T. Yamada, On the Diophantine equation x m =y n 1 +y n 2 ++y n k , Glasg. Math. J. 51 (2009), 143–148. | MR 2471683 | Zbl 1217.11036