Lucas' Theorem modulo pn via Harmonic sums
Abstract
Lucas' theorem states that for a prime p, integers a,b≥0 and 0≤ r,s≤ p−1, one has binompa+rpb+s≡binomabbinomrsbmod p. Let D(pn) be the set of pairs (r,s) with s≤ r for which this congruence holds modulo pn for all a,b≥0. Rowland characterized D(p2) in terms of harmonic sums and symmetries of Pascal's triangle. For n<p, we give a criterion for a point (r,s) to lie in D(pn) in terms of generalized harmonic sums of order at most n−1. We show that D(pn) is invariant under the reflection (r,s)↦(r,r−s) and the rotation (r,s)↦(p−1−s,r−s), so that with every point it contains its full orbit under the dihedral group D3. Applied to the vertices, our criterion gives a result of Jacobsthal-Ljunggren type modulo pn. In particular, we recover Zhao's result (0,0)∈ D(p4) if and only if p is a Wolstenholme prime. We also study the centroid and the edge midpoints, whose membership in D(p3) and D(p4) is governed by Bernoulli numbers and polynomials, Wolstenholme primes and super Wieferich primes. In particular, for p=205129 the centroid lies in D(p3), answering a question of Rowland.
Related
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- Lucas recurrences and Lambert series in coprime double sums
- Proofs of some OEIS conjectures on Wythoff sums, Fibonacci and Lucas words
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BibTeX
@misc{kumar2026lucas,
title = {{Lucas' Theorem modulo $p^n$ via Harmonic sums}},
author = {Suneel Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25248},
eprint = {2610.25248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25248},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Lucas' Theorem modulo p^n via Harmonic sums
AU - Suneel Kumar
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25248
UR - https://arxiv.org/abs/2610.25248
AB - Lucas' theorem states that for a prime p, integers a,b≥0 and 0≤ r,s≤ p−1, one has binompa+rpb+s≡binomabbinomrsbmod p. Let D(p^n) be the set of pairs (r,s) with s≤ r for which this congruence holds modulo p^n for all a,b≥0. Rowland characterized D(p^2) in terms of harmonic sums and symmetries of Pascal's triangle. For n<p, we give a criterion for a point (r,s) to lie in D(p^n) in terms of generalized harmonic sums of order at most n−1. We show that D(p^n) is invariant under the reflection (r,s)↦(r,r−s) and the rotation (r,s)↦(p−1−s,r−s), so that with every point it contains its full orbit under the dihedral group D_3. Applied to the vertices, our criterion gives a result of Jacobsthal-Ljunggren type modulo p^n. In particular, we recover Zhao's result (0,0)∈ D(p^4) if and only if p is a Wolstenholme prime. We also study the centroid and the edge midpoints, whose membership in D(p^3) and D(p^4) is governed by Bernoulli numbers and polynomials, Wolstenholme primes and super Wieferich primes. In particular, for p=205129 the centroid lies in D(p^3), answering a question of Rowland.
ER -