Perfect powers with few digits and S-unit coefficients
Abstract
In this paper, we study two questions concerning perfect powers in S-unit equations and sparse representations of perfect powers. Corvaja-Zannier \cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in N having precisely four non-zero digits in their binary expansion. At first, we prove the finiteness of the set of solutions to the equation yd=1+c1gm1+c2gm2+c3gm3, 0<m1<m2<m3, where ci are S-units for a suitable set S. The proof combines consequences of the Subspace Theorem concerning integer values of analytic series at S-unit points with Roth's theorem for number fields. As a second result, we extend the work of Bennett and Bugeaud \cite{bennettbugead2013perfect} by proving that, for every fixed integer base g≥ 2, there exists a constant q0(g) such that, for every prime q>q0(g), no perfect q-th power admits a representation with five non-zero digits in base g. The proof relies on explicit lower bounds for linear forms in both Archimedean and non-Archimedean logarithms.
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BibTeX
@misc{darsana2026perfect,
title = {{Perfect powers with few digits and $S$-unit coefficients}},
author = {Darsana N and Parvathi S Nair and Sudhansu Sekhar Rout},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08310},
eprint = {2610.08310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08310},
}
RIS
TY - UNPB
TI - Perfect powers with few digits and S-unit coefficients
AU - Darsana N
AU - Parvathi S Nair
AU - Sudhansu Sekhar Rout
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08310
UR - https://arxiv.org/abs/2610.08310
AB - In this paper, we study two questions concerning perfect powers in S-unit equations and sparse representations of perfect powers. Corvaja-Zannier \cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in N having precisely four non-zero digits in their binary expansion. At first, we prove the finiteness of the set of solutions to the equation y^d=1+c_1g^m_1+c_2g^m_2+c_3g^m_3, 0<m_1<m_2<m_3, where c_i are S-units for a suitable set S. The proof combines consequences of the Subspace Theorem concerning integer values of analytic series at S-unit points with Roth's theorem for number fields. As a second result, we extend the work of Bennett and Bugeaud \cite{bennettbugead2013perfect} by proving that, for every fixed integer base g≥ 2, there exists a constant q_0(g) such that, for every prime q>q_0(g), no perfect q-th power admits a representation with five non-zero digits in base g. The proof relies on explicit lower bounds for linear forms in both Archimedean and non-Archimedean logarithms.
ER -