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PreprintCC BY 4.0arXiv

Sharp p-adic Extrema and Least Extremal Rows for Restricted Binomial GCDs

  • John Fairfax-Ball
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Abstract

For integers m≥ 2 and rows N>m divisible by m, consider the restricted binomial greatest common divisor G(N;m)=gcd{binom Nk:0<k<N, m| k}. Fix a prime p with pnmid m, and let rp(m) be the least positive integer r such that m<pr. We prove that the largest possible value of vp(G(N;m)), as N ranges over all admissible rows, is exactly rp(m), and we give a constructive equality row. Attainment is established before the least extremal row Tp(m) is defined. For the special family m=pa+1 with a≥2, we determine that least row exactly: Tp(pa+1)=p3a+1. The proof combines Kummer's carry theorem with an explicit leading-digit witness for the universal upper bound, a multiplicative-order construction for equality, and a separate strict-minimality argument below p3a+1 with a fallback witness for the unique worst leading-digit pattern. The selected-GCD family itself is known in the literature; the relation to earlier results and the limits of the documented literature search are stated explicitly.

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Cite

BibTeXRIS
BibTeX
@misc{fairfaxball2026sharp,
  title = {{Sharp $p$-adic Extrema and Least Extremal Rows for Restricted Binomial GCDs}},
  author = {John Fairfax-Ball},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01328},
  eprint = {2610.01328},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01328},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sharp p-adic Extrema and Least Extremal Rows for Restricted Binomial GCDs
AU  - John Fairfax-Ball
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01328
UR  - https://arxiv.org/abs/2610.01328
AB  - For integers m≥ 2 and rows N>m divisible by m, consider the restricted binomial greatest common divisor G(N;m)=gcd{binom Nk:0<k<N, m| k}. Fix a prime p with pnmid m, and let r_p(m) be the least positive integer r such that m<p^r. We prove that the largest possible value of v_p(G(N;m)), as N ranges over all admissible rows, is exactly r_p(m), and we give a constructive equality row. Attainment is established before the least extremal row T_p(m) is defined. For the special family m=p^a+1 with a≥2, we determine that least row exactly: T_p(p^a+1)=p^3a+1. The proof combines Kummer's carry theorem with an explicit leading-digit witness for the universal upper bound, a multiplicative-order construction for equality, and a separate strict-minimality argument below p^3a+1 with a fallback witness for the unique worst leading-digit pattern. The selected-GCD family itself is known in the literature; the relation to earlier results and the limits of the documented literature search are stated explicitly.
ER  -