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Consecutive non-square non-primitive tuples in finite fields

  • Juncheng Zhou
  • Hongfeng Wu
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Abstract

Let q be an odd prime power and put θq=(φ(q−1))/(q−1). An element of Fq is called non-square non-primitive, or \emph{NSNP}, if it is both a non-square and a non-primitive element. Let ℓ be an odd prime divisor of q−1. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square ℓth powers. Combining this estimate with a finite computation, we prove that θq<4/15 guarantees three consecutive NSNP elements. On the boundary θq=4/15, the only exceptions are q∈{31,61,121}. As a further application, we show that for charFq>3, the inequality θq<8/35 guarantees four consecutive NSNP elements, with q=211 as the unique exception on the boundary θq=8/35. Thus both constants are best possible.

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Cite

BibTeXRIS
BibTeX
@misc{zhou2026consecutive,
  title = {{Consecutive non-square non-primitive tuples in finite fields}},
  author = {Juncheng Zhou and Hongfeng Wu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.17267},
  eprint = {2607.17267},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.17267},
}
RIS
TY  - UNPB
TI  - Consecutive non-square non-primitive tuples in finite fields
AU  - Juncheng Zhou
AU  - Hongfeng Wu
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.17267
UR  - https://arxiv.org/abs/2607.17267
AB  - Let q be an odd prime power and put θ_q=(φ(q−1))/(q−1). An element of Fq is called non-square non-primitive, or \emph{NSNP}, if it is both a non-square and a non-primitive element. Let ℓ be an odd prime divisor of q−1. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square ℓth powers. Combining this estimate with a finite computation, we prove that θ_q<4/15 guarantees three consecutive NSNP elements. On the boundary θ_q=4/15, the only exceptions are q∈{31,61,121}. As a further application, we show that for charFq>3, the inequality θ_q<8/35 guarantees four consecutive NSNP elements, with q=211 as the unique exception on the boundary θ_q=8/35. Thus both constants are best possible.
ER  -