Primes with Restricted-Digit Differences

Abstract

Fix a base b≥2 and a digit set 𝒟⊂{0,1,…,b−1}, and let 𝒞_k be the set of integers in [0,b^k) whose k base-b digits all lie in 𝒟. We study prime pairs and three-term arithmetic progressions in primes whose difference, or common difference, belongs to 𝒞_k. Under a natural digit non-resonance condition, we establish a localized Fourier criterion guaranteeing asymptotic formulae with explicit leading constants for both problems. Using a transfer-operator argument, we verify this criterion in explicit parameter ranges for several natural families of digit sets.

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BibTeX RIS

BibTeX

@misc{han2026primes,
  title = {{Primes with Restricted-Digit Differences}},
  author = {Rui Han and Yaghoub Rahimi and Fan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05810},
  eprint = {2609.05810},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05810},
}

RIS

TY  - UNPB
TI  - Primes with Restricted-Digit Differences
AU  - Rui Han
AU  - Yaghoub Rahimi
AU  - Fan Yang
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.05810
UR  - https://arxiv.org/abs/2609.05810
AB  - Fix a base b≥2 and a digit set 𝒟⊂{0,1,…,b−1}, and let 𝒞_k be the set of integers in [0,b^k) whose k base-b digits all lie in 𝒟. We study prime pairs and three-term arithmetic progressions in primes whose difference, or common difference, belongs to 𝒞_k. Under a natural digit non-resonance condition, we establish a localized Fourier criterion guaranteeing asymptotic formulae with explicit leading constants for both problems. Using a transfer-operator argument, we verify this criterion in explicit parameter ranges for several natural families of digit sets.
ER  -

Abstract

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