Pith. sign in

REVIEW 3 minor 14 references

For n = q^t p the binomial congruence binom(qn, n) ≡ q^n mod n holds exactly when a congruence modulo p and a base-q digit sum inequality are both satisfied.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-06-30 05:05 UTC pith:YW3J7DDY

load-bearing objection Reduces the search in the q^t p family to a mod p condition plus digit sum check, and ties squares to Wieferich primes.

arxiv 2606.30232 v1 pith:YW3J7DDY submitted 2026-06-29 math.NT math.CO

Structured Solutions of Prime-Base Binomial Congruences

Gabriel Ara\'ujo Guedes , Ricardo Nunes Machado Junior This is my paper
classification math.NT math.CO
keywords binomial congruencedigit sumWieferich primeprime basemodular conditionfactorizationnumber theoretic family
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the congruence binom(qn, n) ≡ q^n mod n where q is prime. It restricts attention to the family n = q^t p with p a distinct prime. The central result is that the congruence holds precisely when two independent conditions are met: a specific congruence involving p and an inequality on the sum of the base-q digits of n. This separation converts the search for solutions into the concrete steps of factoring one explicit integer and applying a digit-sum filter. The same criterion yields new solutions for several small q and shows that all square solutions n = p^2 correspond to Wieferich primes in base q.

Core claim

For n belonging to the family q^t p with p a prime different from q, the congruence binom(qn, n) ≡ q^n mod n holds if and only if a certain congruence condition is satisfied modulo p and the sum of the digits of n in base q satisfies a given inequality. This equivalence turns the search for solutions into the tasks of factoring an explicit integer and checking the digit sum condition. Separately, the square case n = p^2 is solved exactly by the primes p that are Wieferich primes to the base q.

What carries the argument

The equivalence of the binomial congruence to an independent modular condition modulo p together with a base-q digit-sum inequality, restricted to the family n = q^t p.

Load-bearing premise

The claimed equivalence between the binomial congruence and the two conditions applies only inside the restricted family of n equal to q to a power times one other distinct prime.

What would settle it

Take any specific n of the form q^t p, compute both sides of the congruence directly, and check whether the two conditions hold; disagreement between the congruence holding and the conditions holding would falsify the equivalence.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper studies the congruence inom{qn}{n} ≡ q^n mod n for prime base q. It restricts attention to the family n = q^t p (p prime, p ≠ q) and claims that membership in this family reduces the original congruence to the conjunction of an independent congruence modulo p and a base-q digit-sum inequality. The resulting criterion is applied to produce new explicit solutions for q in {2,3,5,7,11}; separately, the square case n = p^2 is shown to be governed exactly by Wieferich primes to base q.

Significance. If the stated equivalence is correct, the work supplies an explicit, factor-and-filter search procedure that is directly applicable inside the chosen family and yields concrete new solutions. The reduction to a digit-sum test and the clean characterization of the square case via Wieferich primes are both useful; the explicit link to OEIS A080469 is also noted. The scope is deliberately limited to n = q^t p, so the result does not claim to resolve the general problem.

minor comments (3)
  1. The abstract and introduction should state the precise range of t (t ≥ 1 or t ≥ 0) for which the equivalence is proved; this affects the digit-sum inequality and should be fixed in §1.
  2. Notation for the base-q digit sum s_q(·) is introduced without an explicit definition; add a short sentence or displayed equation in the preliminaries.
  3. The computational examples in the tables would benefit from an additional column listing the factored value of the auxiliary integer whose prime factors are tested, to make the search procedure fully reproducible from the text alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report, so we have no specific points to address point-by-point. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives a logical equivalence, scoped explicitly to the family n = q^t p (p prime ≠ q), between the binomial congruence and the conjunction of a single congruence modulo p plus a base-q digit-sum inequality. This equivalence is obtained by direct analysis of the restricted family rather than by fitting parameters to data, renaming known results, or reducing via self-citation chains. The square case n = p^2 is characterized using the external notion of Wieferich primes. No step reduces by construction to its own inputs; the derivation chain remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts about binomial coefficients modulo primes and base-q digit sums; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of binomial coefficients and modular arithmetic (including behavior modulo primes)
    Invoked to establish the equivalence between the original congruence and the two simpler conditions.

pith-pipeline@v0.9.1-grok · 5695 in / 1351 out tokens · 47218 ms · 2026-06-30T05:05:53.015269+00:00 · methodology

0 comments
read the original abstract

In this paper, we study the congruence $\binom{qn}{n} \equiv q^n \pmod n$ for a prime base $q$. Motivated by the OEIS sequence \seqnum{A080469} and the conjectural existence of infinitely many ternary solutions of the form $n=3^t p$, we analyze the more general family $n=q^t p$, where $p\neq q$ is prime. Our main result shows that, in this family, the congruence is equivalent to two independent conditions: a congruence modulo $p$ and an inequality in the sum of the digits. This reduces the search for such solutions to factoring an explicit integer and applying a base-$q$ digit-sum filter. We use this criterion to produce new large solutions for $q\in\{2,3,5,7,11\}$. We also prove that square solutions $n=p^2$ are exactly governed by Wieferich primes in base $q$.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Alpern,Generic Integer Factorization Calculator, online web app and GitHub repository, 2026

    D. Alpern,Generic Integer Factorization Calculator, online web app and GitHub repository, 2026. Available athttps: //www.alpertron.com.ar/ECMC.HTM. Source code available athttps://github.com/alpertron/calculators

    2026

  2. [2]

    Ballot, Another Lucasnomial generalization of Wolstenholme’s congruence,Journal of Integer Sequences23(2020), Article 20.3.7

    C. Ballot, Another Lucasnomial generalization of Wolstenholme’s congruence,Journal of Integer Sequences23(2020), Article 20.3.7

    2020

  3. [3]

    K. S. Davis and W. A. Webb, Lucas’ theorem for prime powers,European Journal of Combinatorics11(1990), 229–233

    1990

  4. [4]

    F. G. Dorais and D. Klyve, A Wieferich prime search up to 6.7·10 15,Journal of Integer Sequences14(2011), Article 11.9.2

    2011

  5. [5]

    Granville, Arithmetic properties of binomial coefficients I: Binomial coefficients modulo prime powers, inOrganic Mathematics, CMS Conference Proceedings, Vol

    A. Granville, Arithmetic properties of binomial coefficients I: Binomial coefficients modulo prime powers, inOrganic Mathematics, CMS Conference Proceedings, Vol. 20, American Mathematical Society, 1997, pp. 253–275. Available at https://www.cecm.sfu.ca/organics/papers/granville/paper/binomial/html/binomial.html

    1997

  6. [6]

    Guedes and R

    G. Guedes and R. Machado, Perfect numbers, Wieferich primes and the solutions of 2n n ≡2 n (modn),Notes on Number Theory and Discrete Mathematics29(2023), 705–712. Available athttps://nntdm.net/papers/nntdm-29/ NNTDM-29-4-705-712.pdf

    2023

  7. [7]

    Helou and G

    C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse,Journal of Number Theory128(2008), 475–499

    2008

  8. [8]

    N. M. Katz, Wieferich past and future, survey note, 2012. Available athttps://web.math.princeton.edu/ ~nmk/ wieferich42.pdf

    2012

  9. [9]

    Keller and J

    W. Keller and J. Richstein, Solutions of the congruencea p−1 ≡1 (modp r),Mathematics of Computation74(2005), 927–936

    2005

  10. [10]

    Machado and G

    R. Machado and G. Guedes,Congruences on primeq, GitHub repository. Available athttps://github.com/ prof-ricardomachado/congruences-on-prime-q

  11. [11]

    Martin,Primo: Elliptic Curve Primality Proving software

    M. Martin,Primo: Elliptic Curve Primality Proving software. Available athttp://www.ellipsa.eu/public/primo/primo. html

  12. [12]

    R. J. McIntosh, On the converse of Wolstenholme’s theorem,Acta Arithmetica71(1995), 381–389

    1995

  13. [13]

    Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

    R. Meˇ strovi´ c, Wolstenholme’s theorem: Its generalizations and extensions in the last hundred and fifty years (1862–2012), preprint, 2011. Available athttps://arxiv.org/abs/1111.3057

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  14. [14]

    Meˇ strovi´ c, On a congruence modulon3 involving two consecutive sums of powers,Journal of Integer Sequences17 (2014), Article 14.8.4

    R. Meˇ strovi´ c, On a congruence modulon3 involving two consecutive sums of powers,Journal of Integer Sequences17 (2014), Article 14.8.4. Departamento de Matem´atica, Universidade Federal Rural de Pernambuco, Recife, PE 52171-900, Brasil Email address:gabriel.guedes@ufrpe.br Departamento de Matem´atica, Universidade Federal Rural de Pernambuco, Recife, P...

    2014