# 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

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topCorvaja, 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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