Restricted Binomial GCDs at Primes Congruent to -1
Abstract
For integers m≥ 2 and m| N, let G(N;m)=gcd{binomNk:0<k<N, m| k}. We prove a complete p-adic valuation formula for G(N;m) at primes p≡ −1pmod m, under the hypotheses m≥ 3, m| N, and m<N. Writing N=∑i di pi, put A=∑i evendi and B=∑i odddi. Then vp(G(N;m)) is $2$ in the exceptional mixed case (A,B)=(1,1) with p=m−1, is $1$ in the mixed case (A,B)=(1,1) with m<p, is $1$ in the one-parity cases (A,B)=(m,0) and (A,B)=(0,m), and is $0$ otherwise. The proof uses Kummer's theorem to translate the problem into digitwise borrow counts and a minimal signed zero-sum classification. A source-level literature audit through 25 September 2026 located no equivalent prior theorem for the full minus-one classification: McTague's corrected same-residue extension covers the one-parity p>m subcases, but not the mixed-parity branch or the p=m−1 regime. The theorem, its plus-one companion, a scaling reduction, and the m=3,4,6 specializations have been formalized in Lean 4 / Mathlib.
Related
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- Primes with Restricted-Digit Differences
- Congruent quotient matrix, nodal domain, and nodal count
- Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields
- Tilings of Closed Surfaces by Congruent Most Curvilinear Polygons
BibTeX
@misc{fairfaxball2026restricted,
title = {{Restricted Binomial GCDs at Primes Congruent to -1}},
author = {John Fairfax-Ball},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37754},
eprint = {2609.37754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37754},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Restricted Binomial GCDs at Primes Congruent to -1
AU - John Fairfax-Ball
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37754
UR - https://arxiv.org/abs/2609.37754
AB - For integers m≥ 2 and m| N, let G(N;m)=gcd{binomNk:0<k<N, m| k}. We prove a complete p-adic valuation formula for G(N;m) at primes p≡ −1pmod m, under the hypotheses m≥ 3, m| N, and m<N. Writing N=∑_i d_i p^i, put A=∑_i evend_i and B=∑_i oddd_i. Then v_p(G(N;m)) is $2$ in the exceptional mixed case (A,B)=(1,1) with p=m−1, is $1$ in the mixed case (A,B)=(1,1) with m<p, is $1$ in the one-parity cases (A,B)=(m,0) and (A,B)=(0,m), and is $0$ otherwise. The proof uses Kummer's theorem to translate the problem into digitwise borrow counts and a minimal signed zero-sum classification. A source-level literature audit through 25 September 2026 located no equivalent prior theorem for the full minus-one classification: McTague's corrected same-residue extension covers the one-parity p>m subcases, but not the mixed-parity branch or the p=m−1 regime. The theorem, its plus-one companion, a scaling reduction, and the m=3,4,6 specializations have been formalized in Lean 4 / Mathlib.
ER -