PreprintCC BY 4.0arXiv
The Strong 24-Conjecture
Abstract
Restricted forms of Lagrange's four-square theorem, in which the representing variables are required to satisfy additional arithmetic conditions, have been studied from several points of view. Among the refinements proposed by Zhi-Wei Sun is the conjecture that every nonnegative integer has a representation N=x2+y2+z2+w2, x,y,z,w∈ℤ≥0, for which x and x+24y are perfect squares. We prove this conjecture for all sufficiently large integers. The proof starts from an elementary parametrization of the two square conditions and combines a semi-linear two-squares sieve with Poisson summation, finite-field cancellation, and switching.
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BibTeX
@misc{he2026strong,
title = {{The Strong 24-Conjecture}},
author = {Bo He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13338},
eprint = {2609.13338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13338},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Strong 24-Conjecture
AU - Bo He
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13338
UR - https://arxiv.org/abs/2609.13338
AB - Restricted forms of Lagrange's four-square theorem, in which the representing variables are required to satisfy additional arithmetic conditions, have been studied from several points of view. Among the refinements proposed by Zhi-Wei Sun is the conjecture that every nonnegative integer has a representation N=x^2+y^2+z^2+w^2, x,y,z,w∈ℤ_≥0, for which x and x+24y are perfect squares. We prove this conjecture for all sufficiently large integers. The proof starts from an elementary parametrization of the two square conditions and combines a semi-linear two-squares sieve with Poisson summation, finite-field cancellation, and switching.
ER -