Finiteness of odd perfect powers with four nonzero binary digits

Pietro Corvaja[1]; Umberto Zannier[2]

  • [1] Dipartimento di Matematica e Informatica Via delle Scienze, 206 33100 Udine (Italy)
  • [2] Scuola Normale Superiore Piazza dei Cavalieri, 7 56100 Pisa (Italy)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 2, page 715-731
  • ISSN: 0373-0956

Abstract

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We prove that there are only finitely many odd perfect powers in ℕ having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at S -unit points (in a suitable ν -adic convergence), Roth’s general theorem, 2 -adic Padé approximations (by integers) to numbers in varying number fields and lower bounds for linear forms in two logarithms (both in the usual and in the 2 -adic context).

How to cite

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Corvaja, Pietro, and Zannier, Umberto. "Finiteness of odd perfect powers with four nonzero binary digits." Annales de l’institut Fourier 63.2 (2013): 715-731. <http://eudml.org/doc/275565>.

@article{Corvaja2013,
abstract = {We prove that there are only finitely many odd perfect powers in $\mathbb\{N\}$ having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at $S$-unit points (in a suitable $\nu $-adic convergence), Roth’s general theorem, $2$-adic Padé approximations (by integers) to numbers in varying number fields and lower bounds for linear forms in two logarithms (both in the usual and in the $2$-adic context).},
affiliation = {Dipartimento di Matematica e Informatica Via delle Scienze, 206 33100 Udine (Italy); Scuola Normale Superiore Piazza dei Cavalieri, 7 56100 Pisa (Italy)},
author = {Corvaja, Pietro, Zannier, Umberto},
journal = {Annales de l’institut Fourier},
keywords = {Diophantine equations; diophantine approximations; perfect powers; exponential diophantine equation; diophantine approximation; perfect power},
language = {eng},
number = {2},
pages = {715-731},
publisher = {Association des Annales de l’institut Fourier},
title = {Finiteness of odd perfect powers with four nonzero binary digits},
url = {http://eudml.org/doc/275565},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Corvaja, Pietro
AU - Zannier, Umberto
TI - Finiteness of odd perfect powers with four nonzero binary digits
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 2
SP - 715
EP - 731
AB - We prove that there are only finitely many odd perfect powers in $\mathbb{N}$ having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at $S$-unit points (in a suitable $\nu $-adic convergence), Roth’s general theorem, $2$-adic Padé approximations (by integers) to numbers in varying number fields and lower bounds for linear forms in two logarithms (both in the usual and in the $2$-adic context).
LA - eng
KW - Diophantine equations; diophantine approximations; perfect powers; exponential diophantine equation; diophantine approximation; perfect power
UR - http://eudml.org/doc/275565
ER -

References

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  7. A. Schinzel, Polynomials with special regard to reducibility, (2000), Cambridge University Press Zbl0956.12001MR1770638
  8. M. Waldschmidt, Linear Independence Measures for Logarithms of Algebraic Numbers, Diophantine approximation 1819 (2003), 249-344, AmorosoF.F. Zbl1093.11054MR2009832
  9. K. Yu, p -adic logarithmic forms and group varieties. II, Acta Arith. 89 (1999), 337-378 Zbl0928.11031MR1703864
  10. U. Zannier, Roth Theorem, Integral Points and certain ramified covers of ℙ 1 , Analytic Number Theory - Essays in Honour of Klaus Roth (2009), 471-491, Cambridge University Press Zbl1231.11069MR2508664

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