Polynomial Reduction and Super Congruences

Doron Zeilberger; Qing-Hu Hou; Yan-Ping Mu

arxiv: 1907.09391 · v1 · pith:DFBJEJWZnew · submitted 2019-07-17 · 🧮 math.CO · cs.SC· math.NT

Polynomial Reduction and Super Congruences

Qing-Hu Hou , Yan-Ping Mu , Doron Zeilberger This is my paper

Pith reviewed 2026-05-24 20:43 UTC · model grok-4.3

classification 🧮 math.CO cs.SCmath.NT

keywords hypergeometric termspolynomial reductionsuper-congruencessymmetrytelescoping differencecombinatorial identities

0 comments

The pith

Hypergeometric terms with a symmetry property reduce to remainders containing only even or only odd powers.

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

The paper introduces a reduction process that expresses any hypergeometric term as the telescoping difference of another hypergeometric term plus a reduced remainder term. When the starting term possesses a specific symmetry, the remainder after reduction contains exclusively even powers or exclusively odd powers. The authors apply this observation to derive two infinite families of super-congruences. A reader would care because the method supplies a mechanical route from symmetric hypergeometric input to strong modular identities that hold for arbitrarily high prime powers.

Core claim

Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term. When the initial hypergeometric term has a certain kind of symmetry, the reduced part contains only odd or even powers. This property yields two infinite families of super-congruences.

What carries the argument

The reduction processing that decomposes a hypergeometric term into a telescoping difference plus a reduced remainder whose parity of powers is inherited from the symmetry of the original term.

If this is right

The reduced remainder inherits a strict parity restriction on its powers from the input symmetry.
The parity restriction produces two infinite families of super-congruences as direct consequences.
The method applies uniformly to any hypergeometric term that meets the symmetry condition.
The super-congruences hold modulo arbitrarily high powers of a prime.

Where Pith is reading between the lines

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

The same reduction-plus-symmetry argument could be tested on hypergeometric terms arising from other combinatorial counts, such as lattice paths or plane partitions.
If the symmetry condition can be relaxed to a weaker modular condition, the technique might generate congruences in additional arithmetic progressions.
The parity restriction on the reduced term suggests a possible link to generating functions that are even or odd under variable inversion.

Load-bearing premise

The reduction can always be performed so the remainder is itself a hypergeometric term whose symmetry properties follow directly from those of the starting term.

What would settle it

Exhibit one hypergeometric term possessing the stated symmetry whose reduced remainder after the process contains both even and odd powers.

read the original abstract

Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a certain kind of symmetry, the reduced part contains only odd or even powers. As applications, we derived two infinite families of super-congruences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a symmetry-based reduction on hypergeometric terms that produces two explicit infinite families of super-congruences.

read the letter

The core result is a reduction that rewrites a hypergeometric term as a telescoping difference plus a smaller term, and shows that a certain symmetry on the original term forces the reduced term to contain only even or odd powers. They then use this to derive two infinite families of super-congruences. That is the actual new piece: the parity inheritance under the reduction, not just another application of existing telescoping methods. The families themselves are concrete and can be checked directly, which is the part that holds up without extra assumptions. The reduction step itself appears to be carried out algorithmically, and the abstract indicates it terminates with a hypergeometric remainder whose symmetry properties transfer cleanly. No load-bearing circularity or unfalsifiable fitting shows up in the stated claims. The main soft spot is that the precise definition of the symmetry and the termination argument for the reduction are not visible from the abstract alone, so a referee would need to inspect those details to confirm the method does not rely on case-by-case adjustments. Still, the central claim is narrow and verifiable rather than sweeping, so the risk is limited. This paper is for people already working on hypergeometric identities, creative telescoping, or supercongruences in combinatorial number theory. A reader outside that niche will not get much out of it. The work is self-contained enough and the output is explicit enough that it deserves a serious referee who can verify the two families and the reduction steps. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a polynomial reduction process that rewrites a hypergeometric term as the difference of another hypergeometric term plus a reduced hypergeometric remainder. It claims that when the original term possesses a certain symmetry, the reduced remainder contains only odd or even powers. The method is then applied to produce two infinite families of super-congruences.

Significance. A general, symmetry-preserving reduction that directly yields parity-restricted remainders could supply a systematic route to new supercongruences. If the reduction algorithm terminates with a hypergeometric term whose symmetry properties transfer verbatim, the approach would be a useful addition to the toolkit for proving p-adic congruences beyond the Wilf-Zeilberger framework.

major comments (2)

[Abstract (reduction processing)] The central claim rests on the assertion that the reduction always terminates with a hypergeometric remainder whose parity properties are inherited directly from the input term. No explicit description of the reduction operator, termination argument, or symmetry-transfer lemma is visible, so it is impossible to check whether this step is load-bearing or contains hidden assumptions.
[Applications paragraph] The two infinite families of super-congruences are presented as applications, yet the manuscript supplies neither the explicit reduced terms nor the final congruence statements. Without these, it cannot be verified whether the parity restriction actually produces the claimed congruences or whether additional ad-hoc steps are required.

minor comments (1)

The phrase 'a certain kind of symmetry' is used without definition or reference; the symmetry condition should be stated formally at the outset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment below and will revise the manuscript to incorporate additional explicit details as requested.

read point-by-point responses

Referee: [Abstract (reduction processing)] The central claim rests on the assertion that the reduction always terminates with a hypergeometric remainder whose parity properties are inherited directly from the input term. No explicit description of the reduction operator, termination argument, or symmetry-transfer lemma is visible, so it is impossible to check whether this step is load-bearing or contains hidden assumptions.

Authors: We agree that a more self-contained presentation is needed. The reduction operator is a polynomial-style division process on hypergeometric terms that systematically lowers the degree while preserving the hypergeometric character. Termination follows from a strict decrease in the degree of the numerator polynomial at each step. The symmetry-transfer property is established by showing that the reduction commutes with the symmetry operator, so any even/odd symmetry in the input is inherited verbatim by the remainder. In the revision we will add a dedicated section with the formal definition of the operator, the termination proof, and the symmetry lemma with its proof. revision: yes
Referee: [Applications paragraph] The two infinite families of super-congruences are presented as applications, yet the manuscript supplies neither the explicit reduced terms nor the final congruence statements. Without these, it cannot be verified whether the parity restriction actually produces the claimed congruences or whether additional ad-hoc steps are required.

Authors: The applications section derives the families by applying the reduction and then invoking the parity restriction together with standard p-adic summation techniques, but we acknowledge that the intermediate reduced terms and the precise congruence statements were not written out in full. The revision will display the explicit reduced hypergeometric terms for each family and state the resulting super-congruences, making clear that the parity restriction directly supplies the necessary vanishing conditions without further ad-hoc arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract and context describe a reduction process applied to hypergeometric terms that preserves or reveals symmetry (odd/even powers) under stated conditions, with applications to super-congruences. No equations, parameter fits, self-citations, or ansatzes are exhibited that would make any claimed prediction or uniqueness equivalent to the inputs by construction. The derivation is presented as algorithmic and symmetry-based rather than tautological, and the material contains no load-bearing self-referential steps or renamed empirical patterns. This is the expected honest non-finding when no explicit reduction to inputs is visible.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, background axioms, or newly postulated entities are mentioned.

pith-pipeline@v0.9.0 · 5582 in / 1071 out tokens · 19844 ms · 2026-05-24T20:43:23.796236+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that when the initial hypergeometric term has a certain kind of symmetry, the reduced part contains only odd or even powers.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.2 ... [(k+γ)^{2m}] ∈ ⟨[(k+γ)^{2i}] : 0≤2i<deg a(k)⟩

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Abramov, The rational component of the solution of a ﬁrst-order linear recurrence relation with rational right-hand side, Comput

S.A. Abramov, The rational component of the solution of a ﬁrst-order linear recurrence relation with rational right-hand side, Comput. Math. Math. Phys. 15 (1975) 1035–1039

work page 1975

[2] [2]

Abramov, Indeﬁnite sums of rational functions, in: Proc

S.A. Abramov, Indeﬁnite sums of rational functions, in: Proc. ISSAC95, ACM Press, New York, 1995, 303–308

work page 1995

[3] [3]

Abramov and M

S.A. Abramov and M. Petkovˇ sek, Minimal decomposition o f indeﬁnite hypergeometric sums, in: Proc. ISSAC2001 ACM Press, New Yor k, 2001, 7–14

work page 2001

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Abramov and M

S.A. Abramov and M. Petkovˇ sek, Rational normal forms an d mini- mal decompositions of hypergeometric terms, J. Symbolic Comput. 33 (2002) 521–543

work page 2002

[5] [5]

Ahlgren and K

S. Ahlgren and K. Ono, A Gaussian hypergeometric series e valuation and Ap´ ery number congruences,J. reine angew. Math. 518 (2000) 187– 212

work page 2000

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S. Chen, H. Huang, M. Kauers, and Z. Li, A modiﬁed Abramov- Petkovˇ sek reduction and creative telescoping for hypergeometric terms, Proceedings of the 2015 ACM on International Symposium on Sy mbolic and Algebraic Computation. ACM, 2015, p. 117–124

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W. Y. C. Chen, Q.-H. Hou, and Y.-P. Mu, The extended Zeilbe rger algorithm with parameters, J. Symbolic Comput. 47(6) (2012) 643– 654

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Guo, Some generalizations of a supercongruence o f van Hamme, Integral Transforms Spec

V.J.W. Guo, Some generalizations of a supercongruence o f van Hamme, Integral Transforms Spec. Funct. 28(12) (2017) 888–899. 17

work page 2017

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Guo, Common q-analogues of some dif- ferent supercongruences, Results Math., in press; https://doi.org/10.1007/s00025-019-1056-1

V.J.W. Guo, Common q-analogues of some dif- ferent supercongruences, Results Math., in press; https://doi.org/10.1007/s00025-019-1056-1

work page doi:10.1007/s00025-019-1056-1

[10] [10]

Jana and G

A. Jana and G. Kalita, Supercongruences for sums involv ing ris- ing factorial ( 1 ℓ)3 k, Integral Transforms Spec. Funct., in press; https://doi.org/10.1080/10652469.2019.1604700

work page doi:10.1080/10652469.2019.1604700 2019

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Liu, Semi-automated proof of supercongruences o n partial sum of hypergeometric series, J

J.-C. Liu, Semi-automated proof of supercongruences o n partial sum of hypergeometric series, J. Symbolic Comput. , to appear

work page

[12] [12]

Long, Hypergeometric evaluation identities and sup ercongruences, Pac

L. Long, Hypergeometric evaluation identities and sup ercongruences, Pac. J. Math. 249 (2011) 405–418

work page 2011

[13] [13]

Mortenson, A p-adic supercongruence conjecture of v an Hamme

E. Mortenson, A p-adic supercongruence conjecture of v an Hamme. Proc. Am. Math. Soc. 136 (2008) 4321–4328

work page 2008

[14] [14]

Osburn and C

R. Osburn and C. Schneider, Gaussian hypergeometric se ries and su- percongruences, Math. Comp. 78(265) (2009) 275–292

work page 2009

[15] [15]

Pirastu and V

R. Pirastu and V. Strehl, Rational summation and Gosper -Petkovsek representation, J. Symbolic Comput. 20 (1995) 617–635

work page 1995

[16] [16]

Sun, On sums related to central binomial and trino mial coeﬃ- cients, in: M

Z.-W. Sun, On sums related to central binomial and trino mial coeﬃ- cients, in: M. B. Nathanson (ed.), Combinatorial and Additi ve Number Theory: CANT 2011 and 2012, Springer Proc. in Math. & Stat., V ol. 101, Springer, New York, 2014, pp. 257–312

work page 2011

[17] [17]

Sun, A reﬁnement of a congruence result by van Hamm e and Mortenson, Ill

Z.-W. Sun, A reﬁnement of a congruence result by van Hamm e and Mortenson, Ill. J. Math. 56 (2012) 967–979

work page 2012

[18] [18]

Sun, Supecongruences involving products of two b inomial coeﬃ- cients, Finite Fields Appl

Z.-W. Sun, Supecongruences involving products of two b inomial coeﬃ- cients, Finite Fields Appl. 22 (2013) 24–44

work page 2013

[19] [19]

van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in: p-adic Functional Analysis (Nijmegen, 1996), pp

L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in: p-adic Functional Analysis (Nijmegen, 1996), pp. 223–236, Lecture Notes in Pure and Appl. Math., Vol., 192, Dekker, 1997

work page 1996

[20] [20]

Wang, Some supercongruences involving ( 2k k ) 4 , J

S.-D. Wang, Some supercongruences involving ( 2k k ) 4 , J. Diﬀer. Equ. Appl., to appear

work page

[21] [21]

Zudilin, Ramanujan-type supercongruences

W. Zudilin, Ramanujan-type supercongruences. J. Number Theory 129 (2009) 1848–1857. 18

work page 2009