PreprintarXiv licencearXiv
Improved exponent bounds for Erdős--Selfridge curves
Abstract
Given integers k,ℓ ≥ 2, define the Erdős--Selfridge curve 𝒞k,ℓ : yℓ = x(x+1)⋯ (x+k−1). Bennett and Siksek proved that if (x,y) is a rational point on 𝒞k,ℓ with ℓ a prime and y ≠ 0, then ℓ is bounded (doubly-exponentially) in terms of k. We improve this bound on ℓ when k is sufficiently large. Our method uses combinatorial and sieve-theoretic arguments, as well as results on solutions to generalized Fermat equations.
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BibTeX
@misc{das2026improved,
title = {{Improved exponent bounds for Erdős--Selfridge curves}},
author = {Pranabesh Das and Kyle Pratt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38076},
eprint = {2609.38076},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38076},
}
RIS
TY - UNPB
TI - Improved exponent bounds for Erdős--Selfridge curves
AU - Pranabesh Das
AU - Kyle Pratt
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38076
UR - https://arxiv.org/abs/2609.38076
AB - Given integers k,ℓ ≥ 2, define the Erdős--Selfridge curve 𝒞_k,ℓ : y^ℓ = x(x+1)⋯ (x+k−1). Bennett and Siksek proved that if (x,y) is a rational point on 𝒞_k,ℓ with ℓ a prime and y ≠ 0, then ℓ is bounded (doubly-exponentially) in terms of k. We improve this bound on ℓ when k is sufficiently large. Our method uses combinatorial and sieve-theoretic arguments, as well as results on solutions to generalized Fermat equations.
ER -