A new kind of numbers and related congruences

Zhi-Wei Sun

REVIEW 2 major objections 6 minor 18 references

For every prime p, the Domb-number harmonic sum has an exact p-adic formula, via a new unifying number family.

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2026-08-02 08:02 UTC pith:C7J2D7NM

load-bearing objection Solid paper: new family S_l^(m), q-analogue, and a proof of a 2019 Domb conjecture; the only real dependency is a published, checkable lemma. the 2 major comments →

arxiv 2607.07638 v3 pith:C7J2D7NM submitted 2026-07-08 math.NT math.CO

A new kind of numbers and related congruences

Zhi-Wei Sun This is my paper

classification math.NT math.CO MSC 11B6511A0705A1005A1911B6811B83

keywords multinomial coefficientsDomb numbersFranel numbersp-adic congruencesBernoulli polynomialsLegendre symbolq-analogueharmonic numbers

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.

This paper introduces a family of numbers S_l^(m)(n) — the sum, over all l-tuples of nonnegative integers adding to n, of the m-th power of the multinomial coefficient — and shows it contains both the Franel numbers and the Domb numbers as special cases. The main general result is a congruence: for any prime p not dividing l+1, a certain alternating harmonic-type sum of these numbers vanishes mod p; a q-analogue is also proved. For the Domb case (l=4, m=2), the paper proves the exact first-order p-adic term: the sum of D(n)/n up to p−1 is congruent, mod p^2, to (p/3)(2/5)p B_{p−2}(1/3), where B is a Bernoulli polynomial and (p/3) is a Legendre symbol. This confirms a conjecture from 2019 and pins down the p-adic behaviour that earlier results could only see mod p.

Core claim

The paper introduces S_l^(m)(n) = sum over k_1+...+k_l=n of the m-th power of the multinomial coefficient, and records the identities S_2^(m)(n)=f_n^(m) (Franel numbers) and S_4^(2)(n)=D(n) (Domb numbers). It then proves Theorem 1.1: for p prime with p∤(l+1), the alternating sum ∑_{n=1}^{p−1} (−1)^{mn}/n^{m−1} S_l^(m)(n) ≡ 0 mod p, together with a q-analogue over cyclotomic polynomials. The centrepiece, Theorem 1.3, establishes the previously conjectured mod p^2 evaluation ∑_{n=1}^{p−1} D(n)/n ≡ (p/3)(2/5)p B_{p−2}(1/3) mod p^2. The proof reduces the Domb sum to coefficients of F(x)^5 and F(x)^3 built from squared factorials, and the final Bernoulli-polynomial term comes from a known congrue

What carries the argument

The central object is the multinomial-power sum S_l^(m)(n) and its generating function F(x) = ∑_{k≥0} x^k/(k!)^m. Theorem 1.1 follows by reading the coefficient of x^p in F(x)^{l+1}, which exactly isolates the sum in (1.10); Wilson's theorem turns each binomial factor into (−1)^n mod p. For Theorem 1.3, using m=2, the paper builds a Wronskian-type polynomial W = F(x)G'(x) − F'(x)G(x) from F and the harmonic-number generating function G, then uses coefficient comparisons to reduce the Domb sum to [x^p]F(x)^3, which is congruent to ∑_{n=1}^{p−1} binom(2n,n)/n^2 mod p. A cited lemma (Lemma 3.1) evaluates that central-binomial sum as half a Bernoulli polynomial, giving the p^2 formula.

Load-bearing premise

The load-bearing premise is the imported Lemma 3.1 — that ∑_{n=1}^{p−1} binom(2n,n)/n^2 ≡ (1/2)(p/3)B_{p−2}(1/3) mod p — which the paper cites as known and only sketches; if that congruence fails, the Domb p^2 formula collapses.

What would settle it

Direct modular computation for a small prime: for p=7, evaluate ∑_{n=1}^{6} D(n)/n modulo 49 and compare with (7/3)(2/5)·7·B_5(1/3) mod 49. A mismatch at any prime p≥7 would disprove Theorem 1.3; matching at several primes would support it. One can generate D(n) independently via the recurrence (n+1)^3D(n+1)=2(2n+1)(5n^2+5n+2)D(n)−64n^3D(n−1).

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

If this is right

The mod p congruence (1.10) holds uniformly for every level and height, so Franel and Domb sums are two instances of a single vanishing pattern.
The Domb congruence mod p^2 is now a theorem; the first-order p-adic deviation is exactly the Bernoulli-polynomial term, not merely zero mod p.
The q-analogue (1.13) is a polynomial statement whose p-th cyclotomic specialization recovers the mod p theorem, giving a q-version for all N, not just primes.
Since S_3^(2)(n) equals the polynomial g_n, the general theorem repackages known polynomial congruences and marks the whole S_l^(m) family as a natural subject for p-adic study.
The proof identifies the central-binomial sum as the engine behind the Domb mod p^2 formula; any sharpening of that lemma would directly sharpen the congruence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same coefficient method may yield mod p^2 formulas for other S_l^(m) sums once the analogous binomial-type sums (for example, trinomial central coefficients) are evaluated p-adically; this is a testable extension the paper does not pursue.
The q-congruence likely implies a family of q-supercongruences when N is specialized to prime powers, following a common progression from q-analogues to supercongruences.
Because D(n) counts 2n-step polygons on the diamond lattice, the result is a p-adic statement about a lattice-walk counting sequence, so connections to p-adic properties of 4-step random-walk moments may follow.
The paper's final conjectures about Domb sums weighted by Lucas sequences suggest deeper multiplicative structure; the method developed here may help approach those congruences.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Solid paper: new family S_l^(m), q-analogue, and a proof of a 2019 Domb conjecture; the only real dependency is a published, checkable lemma.

read the letter

Zhi-Wei Sun's new preprint does three things: introduces the multinomial numbers S_l^(m)(n), proves a mod p congruence for them, gives a q-analogue, and confirms his own 2019 conjecture on Domb numbers modulo p^2. The Domb result, Theorem 1.3, is the main event. It gives the exact p-adic first-order term for sum D(n)/n, matching the predicted Bernoulli polynomial. The proof is a genuine derivation: it reduces the sum to coefficients of F(x)^3 and then to the central binomial sum in Lemma 3.1. I checked the algebraic reductions in Section 3—the identities for [x^p]F^2, the W(x) argument, and the Wolstenholme-based step D(p)≡4 mod p^3—and they hold together. The reduction to Lemma 3.1 is clean, and that lemma is published in J. Number Theory by Mattarei and Tautaso. So the main result does not rest on an unverifiable claim, just on a citation.

The general family S_l^(m) is a natural unification: it contains Franel numbers at l=2 and Domb numbers at l=4,m=2. The mod p congruence and its q-analogue are proved by a short generating-function / q-derivative argument that is transparent. That part is elementary but useful. The paper is honest about what is new: it extends Sun's earlier work, and the q-analogue is a neat addition.

Soft spots, in proportion. The biggest is that Lemma 3.1 is not proved in the paper; Remark 3.1 sketches the route but delegates the hard harmonic-number identity. If Lemma 3.1 is wrong, Theorem 1.3 collapses. But the lemma is in a refereed journal and the sketch is plausible, so this is a dependency, not a flaw. Some deductions in Section 3 are compressed—the jump from (3.5) to (3.6) and the congruence for [x^p]F(x)^2 are quick—but they are checkable and check out. There is no machine-checked formalization, so the proof is only as good as the manual checking. The paper confirms the author's own conjecture and uses his earlier harmonic-number lemma, which is fine but means independent verification of the Domb congruence would be welcome.

Who is this for? People working on p-adic congruences for combinatorial numbers. It is a solid extension of an established program, not a conceptual breakthrough. It deserves a serious referee. I would accept it for peer review.

Referee Report

2 major / 6 minor

Summary. The paper introduces a two-parameter family of multinomial sums S_l^(m)(n), showing that this family unifies Franel numbers and Domb numbers. Theorem 1.1 proves a general mod p congruence for sums of these numbers, Theorem 1.2 gives a q-analogue modulo cyclotomic polynomials, and Theorem 1.3 confirms the author's earlier Conjecture 79(i) for Domb numbers by establishing the mod p^2 congruence with B_{p-2}(1/3). The proofs are based on coefficient extraction from generating functions and Jackson q-derivatives; the Domb congruence is reduced to a cited lemma of Mattarei and Tauraso, Wolstenholme's theorem, and standard harmonic-number identities.

Significance. If correct, the paper gives a genuinely unifying framework for Franel and Domb congruences, a new q-analogue, and a proof of a previously open conjecture with an explicit p-adic first-order term. The generating-function arguments are elementary, transparent, and largely verifiable line by line; the confirmation of the Domb conjecture is a concrete advance. Credit is due for the clean unified construction and for stating the final congruence in a fully explicit form. The main caveats are the paper's dependence on the external Lemma 3.1 and several typographical slips in the proof of Theorem 1.3, none of which appear to affect the final result once corrected.

major comments (2)

[§3, proof of Theorem 1.3] In the second displayed equation after the start of the proof, the factor is written as (p^2/(p-n)) * binom(p,n)^2. The correct factor is p^2 / ((p-n) * binom(p,n)^2), equivalently (p-n)/binom(p-1,n)^2. This is what the subsequent coefficient -n - p(2nH_n - 1) requires. As printed, the equality is false; please correct the placement of the binomial factor in the denominator.
[§3, Lemma 3.1 and Remark 3.1] Theorem 1.3's conclusion is obtained by substituting (3.1), and the Bernoulli-polynomial term enters only through this lemma. The paper cites [8] and gives only a sketch in Remark 3.1. Since this is the load-bearing external input, I recommend either proving Lemma 3.1 in the paper or citing a precise theorem/location in [8], so the reader can verify that the exact constant (1/2) * (p/3) * B_{p-2}(1/3) is being invoked rather than a weaker or different statement.

minor comments (6)

[§3, Eq. (3.8)] The displayed identity has an extra factor 2: sum_{k=1}^{p-1} 1/k^2 = sum_{k=1}^{(p-1)/2}(1/k^2 + 1/(p-k)^2), not twice that. The conclusion [x^p]F(x)^2 ≡ 0 is unaffected.
[References and text] The author of [8] is Roberto Tauraso, not 'Tautaso'; also 'Wolstholme' in the proof of Theorem 1.3 should be 'Wolstenholme'.
[Theorem 1.2] The statement uses n both as a parameter in the definition of S_l^(m)(n;q) and as the summation index in the congruence. Rename one of them to avoid ambiguity.
[§4, Conjecture 4.2] The displayed radicals are garbled in the text; they should presumably read sqrt[n+1]{S_3^(3)(n+1)} / sqrt[n]{S_3^(3)(n)}.
[§3, after (3.7)] To conclude that W(x) - (F(x)^2 - 1)/2 ≡ x^{p+1} P_p(x), one also uses that the constant term of this polynomial is zero. This is true because W(0)=0 and F(0)^2 - 1 = 0, but it should be stated explicitly.
[§2, proof of Theorem 1.1] In the reduction from (2.1) to (1.10), the sign factor (-1)^{2m-1} is not tracked. Since the conclusion is a zero congruence this is harmless, but the displayed chain would be clearer if the sign were included.

Circularity Check

0 steps flagged

No circular reduction: Theorem 1.3 is a genuine derivation from generating functions, external Lemma 3.1, and independent prior lemmas.

full rationale

The claimed congruences are not defined into existence. Theorems 1.1 and 1.2 are proved by coefficient extraction from F(x)^l F'(x) and by Jackson q-derivatives; S_l^{(m)} enters only through the identity [x^n]F(x)^l = S_l^{(m)}(n)/(n!)^m, which is a direct multinomial generating-function identity, not a restatement of the target. For Theorem 1.3, the paper reduces the Domb sum to the coefficient [x^p]F(x)^3 via a chain of exact identities (3.5), (3.9), (3.10), and Wolstenholme's theorem gives [x^p]F(x)^4 ≡ 0 mod p; no fitted constants appear. The final Bernoulli polynomial enters only through Lemma 3.1 (a central-binomial sum, attributed to Mattarei and Tautaso [8]), which is an external, independent result about C(2n,n), not about Domb numbers. The auxiliary use of [12, Lemma 2.3] supplies the independent harmonic-number vanishing Σ H_k/k ≡ 0 mod p; this is a published lemma, and it is not the target congruence. The fact that Theorem 1.3 confirms the author's own earlier Conjecture 79(i) is the statement of the problem being proved, not an input to the proof. Thus there is no self-definitional step, no fitted-input-as-prediction, and no load-bearing self-citation chain. The only notable weakness is the delegation of Lemma 3.1 to [8] with only a sketch in Remark 3.1; that is an external dependency, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No fitted parameters: the proof is analytic and all constants are structural. The new numbers S_l^(m) and their q-analogues are definitions, not postulates used to force the result, so they are not recorded as invented entities. The central Domb conclusion is not built into the assumptions; Lemma 3.1 supplies the Bernoulli polynomial, but it is an external prior result, not a restatement of the target.

axioms (5)

standard math Wilson's theorem: (p−1)! ≡ −1 mod p.
Used in the proof of Theorem 1.1 to evaluate the denominator in the coefficient comparison.
standard math Wolstenholme's theorem: binom(2p−1,p−1) ≡ 1 mod p^3 for p > 3.
Used in the proof of Theorem 1.3 to conclude D(p) ≡ 4 mod p^3.
domain assumption Lemma 3.1 (Mattarei–Tautaso): ∑_{n=1}^{p−1} binom(2n,n)/n^2 ≡ (1/2)(p/3)B_{p−2}(1/3) mod p.
Key external input for Theorem 1.3; cited as known ([8]) with only a sketch given in Remark 3.1.
domain assumption Sun's harmonic-number lemma: ∑_{k=1}^{p−1} H_k/k ≡ 0 mod p (and its proof).
Used in the proof of Theorem 1.3 to show [x^p]W ≡ 0 mod p; a self-cited prior published result.
standard math Jackson q-derivative property: [x^{N−1}] ∂_q P = [N]_q [x^N] P.
Used in the proof of Theorem 1.2 to relate coefficients of the q-generating function.

pith-pipeline@v1.3.0-alltime-deepseek · 9801 in / 33896 out tokens · 264610 ms · 2026-08-02T08:02:24.204477+00:00 · methodology

0 comments

read the original abstract

For integers $l>0$ and $m\geqslant0$, we introduce the numbers $$S_l^{(m)}(n) = \sum_{k_1,\ldots,k_l\in\mathbb N\atop k_1+\cdots+k_l = n} \binom n{k_1,\ldots,k_l}^m \ \ (n=0,1,2,\ldots),$$ and prove that for any prime $p$ not dividing $l+1$ we have the congruence $$\sum_{n=1}^{p-1}\frac{(-1)^{mn}}{n^{m-1}}S_l^{(m)}(n)\equiv0\pmod p.$$ We also obtain a $q$-analogue of this result. For the Domb numbers given by $$D(n)=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}=S_4^{(2)}(n)\ \ (n=0,1,2,\ldots),$$ we confirm a previous conjecture which states that $$\sum_{n=1}^{p-1}\frac{D(n)}n\equiv\left(\frac p3\right)\frac 25pB_{p-2}\left(\frac13\right)\pmod{p^2}$$ for any prime $p$, where $(\frac p3)$ is the Legendre symbol, and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

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Works this paper leans on

18 extracted references

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S. Mattarei and R. Tautaso,Congruences for central binomial sums and finite polylogarithms, J. Number Theory133(2013), 131–157

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Z.-W. Sun,Congruences for Franel numbers, Adv. Appl. Math.51(2013), 524–535

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Z.-W. Sun,Conjectures and results onx 2 modp 2 with4p=x 2 +dy 2, in: Number Theory and Related Area (eds., Y. Ouyang, C. Xing, F. Xu and P. Zhang), Adv. Lect. Math. 27, Higher Education Press and International Press, Beijing-Boston, 2013, pp. 149–197

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Sun,Conjectures involving arithmetical sequences, in: Number Theory: Arithmetic in Shangri-La (eds., S

Z.-W. Sun,Conjectures involving arithmetical sequences, in: Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H. Li and J. Liu), Proc. 6th China-Japan Seminar (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244–258

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Sun,Congruences involvingg n(x) = Pn k=0 n k 22k k xk, Ramanujan J

Z.-W. Sun,Congruences involvingg n(x) = Pn k=0 n k 22k k xk, Ramanujan J. 40(2016), 511–533

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Sun,Open conjectures on congruences, Nanjing Univ

Z.-W. Sun,Open conjectures on congruences, Nanjing Univ. J. Math. Biquar- terly36(2019), no. 1, 1–99

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Sun,New type series for powers ofπ, J

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Wolstenholme,On certain properties of prime numbers, Quart

J. Wolstenholme,On certain properties of prime numbers, Quart. J. Appl. Math.5(1862), 35–39. School of Mathematics, Nanjing University, Nanjing 210093, Peo- ple’s Republic of China Email address:zwsun@nju.edu.cn

[1] [1]

G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge Univ. Press, Cambridge, 1999

1999

[2] [2]

J. M. Campbell,On modular forms and Domb numbers, Integral Transforms Spec. Funct.36(2025), 439–448

2025

[3] [3]

H. H. Chan, S. H. Chan and Z. Liu, Domb’s numbers and Ramanujan-Sato type series for 1/π, Adv. Math.186(2004), 396–410

2004

[4] [4]

Franel,On a question of Laisant, L’Interm´ ediaire des Math´ ematiciens,1 (1894), 45–47

J. Franel,On a question of Laisant, L’Interm´ ediaire des Math´ ematiciens,1 (1894), 45–47

[5] [5]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, New York, 1994

1994

[6] [6]

Guo, G.-S

V.J.W. Guo, G.-S. Mao and H. Pan,Proof of a conjecture involving Sun poly- nomials, J. Difference Equ. Appl.22(2016), 1184–1197

2016

[7] [7]

Liu,Supercongruences for sums involving Domb numbers, Bull

J.-C. Liu,Supercongruences for sums involving Domb numbers, Bull. Sci. Math. 169(2021), Article 102992

2021

[8] [8]

Mattarei and R

S. Mattarei and R. Tautaso,Congruences for central binomial sums and finite polylogarithms, J. Number Theory133(2013), 131–157

2013

[9] [9]

M. D. Rogers,New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for1/π, Ramanujan J.18(2009), 327–340

2009

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Sloane, Sequence A002895 (Domb numbers) at OEIS (On-Line Ency- clopedia of Integer Sequences), https://oeis.org/A002895

N.J.A. Sloane, Sequence A002895 (Domb numbers) at OEIS (On-Line Ency- clopedia of Integer Sequences), https://oeis.org/A002895

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Sun,Congruences for Franel numbers, Adv

Z.-W. Sun,Congruences for Franel numbers, Adv. Appl. Math.51(2013), 524–535

2013

[12] [12]

Sun,Arithmetic theory of harmonic numbers, Proc

Z.-W. Sun,Arithmetic theory of harmonic numbers, Proc. Amer. Math. Soc. 140(2012), 415–428. 16 Z.-W. SUN

2012

[13] [13]

Sun,Conjectures and results onx 2 modp 2 with4p=x 2 +dy 2, in: Number Theory and Related Area (eds., Y

Z.-W. Sun,Conjectures and results onx 2 modp 2 with4p=x 2 +dy 2, in: Number Theory and Related Area (eds., Y. Ouyang, C. Xing, F. Xu and P. Zhang), Adv. Lect. Math. 27, Higher Education Press and International Press, Beijing-Boston, 2013, pp. 149–197

2013

[14] [14]

Sun,Conjectures involving arithmetical sequences, in: Number Theory: Arithmetic in Shangri-La (eds., S

Z.-W. Sun,Conjectures involving arithmetical sequences, in: Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H. Li and J. Liu), Proc. 6th China-Japan Seminar (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244–258

2011

[15] [15]

Sun,Congruences involvingg n(x) = Pn k=0 n k 22k k xk, Ramanujan J

Z.-W. Sun,Congruences involvingg n(x) = Pn k=0 n k 22k k xk, Ramanujan J. 40(2016), 511–533

2016

[16] [16]

Sun,Open conjectures on congruences, Nanjing Univ

Z.-W. Sun,Open conjectures on congruences, Nanjing Univ. J. Math. Biquar- terly36(2019), no. 1, 1–99

2019

[17] [17]

Sun,New type series for powers ofπ, J

Z.-W. Sun,New type series for powers ofπ, J. Comb. Number Theory12 (2020), no. 3, 157–208

2020

[18] [18]

Wolstenholme,On certain properties of prime numbers, Quart

J. Wolstenholme,On certain properties of prime numbers, Quart. J. Appl. Math.5(1862), 35–39. School of Mathematics, Nanjing University, Nanjing 210093, Peo- ple’s Republic of China Email address:zwsun@nju.edu.cn