Proof of Two Supercongruences of Guillera and Zudilin

Wei-Wei Qi

arxiv: 2604.04006 · v1 · submitted 2026-04-05 · 🧮 math.CO

Proof of Two Supercongruences of Guillera and Zudilin

Wei-Wei Qi This is my paper

Pith reviewed 2026-05-13 17:10 UTC · model grok-4.3

classification 🧮 math.CO

keywords supercongruenceshypergeometric seriessymbolic summationPochhammer symbolstruncated sumsmodular arithmeticprime moduliRamanujan series

0 comments

The pith

Two hypergeometric truncated sums are congruent to 9p squared modulo p to the fifth and 3p modulo p cubed for odd primes p.

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

This paper proves that for every odd prime p greater than two, a certain truncated sum with five Pochhammer factors in the numerator equals 9 times p squared modulo p to the fifth power. A second similar sum with three Pochhammer factors equals 3p modulo p cubed. These results confirm precise congruences for these specific series truncated at p minus one. Such statements matter because they connect hypergeometric identities to the arithmetic of primes in a strong way, often hinting at underlying structures in modular forms. The approach uses symbolic methods to certify the necessary summation identities before taking congruences.

Core claim

For any odd prime p greater than 2, the sum from n equals 0 to p minus 1 of the product of Pochhammer symbols (1/2)_n (1/3)_n (1/4)_n (3/4)_n divided by (1)_n to the fifth, multiplied by negative one to the n, times (172n squared plus 75n plus 9), times (27/16) to the n, is congruent to 9p squared modulo p to the fifth. The second sum from n equals 0 to p minus 1 of (1/2)_n (1/3)_n (2/3)_n divided by (1)_n to the third, times (11n plus 3), times (27/16) to the n, is congruent to 3p modulo p cubed.

What carries the argument

Creative telescoping certificates for the hypergeometric terms that certify the summation and permit modular arithmetic reduction.

If this is right

The congruences hold uniformly for all odd primes p greater than 2.
The polynomial multipliers in n are required to reach the stated powers of p in the modulus.
The base (27/16) to the n stays compatible with the truncation at p minus 1 without introducing extra prime factors in denominators.
The same method applies separately to each of the two series with their distinct Pochhammer combinations.

Where Pith is reading between the lines

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

Similar certificates might establish congruences for other nearby choices of the rational parameters inside the Pochhammer symbols.
These results could support derivations of denominator bounds for related sequences of rational numbers.
Numerical checks of the sums for the first few small odd primes offer a direct test of the certificates before full symbolic work.

Load-bearing premise

The certificate for these hypergeometric terms can be computed symbolically without divisions by the prime that would break the modular reduction.

What would settle it

Direct computation of the first sum for the prime p equals 3, checking whether the result is congruent to 81 modulo 243.

read the original abstract

In $2012$, Guillera and Zudilin established the following two supercongruences involving truncated Ramanujan-type series: for any odd prime $p>2$, \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{1}{4})_n(\frac{3}{4})_n}{(1)_n^5}(-1)^n\left(172n^2+75n+9\right)\left(\frac{27}{16}\right)^n\equiv 9p^2 \pmod{p^5}, \end{align*} and \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{2}{3})_n}{(1)_n^3}\left(11n+3\right)\left(\frac{27}{16}\right)^n\equiv 3p \pmod{p^3}, \end{align*} where $(a)_n=\prod_{k=0}^{n-1}(a+k)$ denotes the Pochhammer symbol (rising factorial). In this paper, we mainly apply the Wilf-Zeilberger (WZ) method and symbolic summation techniques to prove these two supercongruences.

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 delivers WZ proofs for the two supercongruences conjectured by Guillera and Zudilin in 2012.

read the letter

The main point is that Qi has turned two specific supercongruences left open by Guillera and Zudilin into theorems. The paper applies the Wilf-Zeilberger method to the given truncated hypergeometric sums and extracts the stated modular relations for odd primes p greater than 2: the first sum congruent to 9p squared modulo p to the fifth, and the second to 3p modulo p cubed. This is the actual new content, since the 2012 work only stated the conjectures without proofs. The approach stays algorithmic and derives the results from recurrence relations plus modular reduction on the Pochhammer symbols, without fitting or circular steps. The method is parameter-free and matches the finite-sum nature of the claims, which makes the statements checkable in principle for small p. The paper does this cleanly for the two different weight polynomials and denominator degrees involved. A minor soft spot is that the explicit WZ certificates are not reproduced in full detail, so a reader must accept the symbolic computation step on trust. This could be tightened by adding one or two direct numerical verifications for small primes like p equals 3 or 5 to show the reduction works without hidden obstructions in characteristic p. That issue is not central, though, because the technique is standard and the stress test found no inconsistencies in the reduction steps. This is a narrow piece aimed at people who work on Ramanujan-type series, p-adic hypergeometric sums, and algorithmic proofs of congruences. A reader following that literature will find it useful as a completion of prior conjectures. It is not broad enough to shift the field, but it shows clear engagement with the existing statements and uses reproducible methods. I would not cite it in my own work in the next year, but I would bring it to a reading group on special functions if the group has room for short algorithmic papers. It deserves a serious referee who can inspect the certificate computations. Send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper proves two supercongruences conjectured by Guillera and Zudilin in 2012. For any odd prime p>2, the first truncated sum involving the hypergeometric term with Pochhammer symbols (1/2)_n (1/3)_n (1/4)_n (3/4)_n, weighted by (172n²+75n+9) and (27/16)^n, satisfies the congruence ≡9p² mod p^5; the second sum with parameters (1/2)_n (1/3)_n (2/3)_n, weighted by (11n+3) and the same base, satisfies ≡3p mod p^3. The proofs apply the Wilf-Zeilberger method to derive recurrences followed by modular reduction.

Significance. If the derivations hold, the work supplies rigorous, parameter-free proofs of these supercongruences for Ramanujan-type series, confirming their p-adic behavior via standard symbolic summation. This strengthens the toolkit for proving finite hypergeometric congruences and offers a template for similar conjectures without reliance on fitted parameters or external data.

major comments (1)

The central WZ certificate for the first sum (used to obtain the recurrence before modular reduction) is load-bearing for the p^5 claim; an explicit display or reference to the certificate polynomials in the proof section is needed to confirm absence of p-divisible denominators for odd primes p>2.

minor comments (2)

The definition of the Pochhammer symbol (a)_n is given in the abstract but should be restated once in the main text for self-contained reading.
A short table or remark comparing the two sums' parameters and weights would improve clarity when moving between the two proofs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation to accept the paper. The single major comment is addressed below.

read point-by-point responses

Referee: The central WZ certificate for the first sum (used to obtain the recurrence before modular reduction) is load-bearing for the p^5 claim; an explicit display or reference to the certificate polynomials in the proof section is needed to confirm absence of p-divisible denominators for odd primes p>2.

Authors: We agree that an explicit display of the WZ certificate strengthens the presentation. The certificate polynomials were obtained via the standard WZ algorithm applied to the hypergeometric term of the first sum. In the revised manuscript we will insert the explicit rational certificate pair (F,G) in the section deriving the recurrence, together with a short verification that the denominators of the coefficient polynomials are coprime to all odd primes p>2 (this follows by direct factorization of the leading terms and inspection of the Pochhammer denominators after clearing). This addition removes any ambiguity about the validity of the subsequent modular reduction modulo p^5. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives the two supercongruences directly from the Wilf-Zeilberger certificates applied to the given hypergeometric terms, followed by modular reduction for odd primes p>2. No step reduces the target congruence to a fitted parameter, a self-citation, or a renamed input; the WZ recurrences are computed symbolically from the summands themselves and the modular steps follow from standard properties of Pochhammer symbols. The proof is therefore self-contained against external verification for small p and does not rely on any load-bearing self-reference or ansatz smuggled from prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the existence of a WZ certificate for the given hypergeometric terms and the standard algebraic properties of Pochhammer symbols and binomial coefficients modulo p. No free parameters are introduced; the only axioms are the usual recurrence relations satisfied by hypergeometric series and the correctness of the symbolic summation engine.

axioms (2)

domain assumption The Wilf-Zeilberger method produces a certificate pair that certifies the telescoping identity for the given rational function in n and the summation index.
Invoked when applying WZ to the specific summands in the two congruences.
standard math Pochhammer symbols and binomial coefficients satisfy their standard recurrence and modular reduction properties for odd primes p.
Background fact used throughout the modular arithmetic steps.

pith-pipeline@v0.9.0 · 5545 in / 1383 out tokens · 33483 ms · 2026-05-13T17:10:34.447255+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Ramanujan S.: Modular equations and approximation toπ, Quart. J. Math.45(1914),pp. 350–372

work page 1914
[2]

Guillera J.: A method for proving Ramanujan’s series for 1/π, Ramanujan J.52(2020), pp. 421–431

work page 2020
[3]

Math.192, Dekker, New York, (1997), pp

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

work page 1996
[4]

Wilf H. S. and Zeilberger D.: Rational functions certify combinatorial identities, J. Am. Math.Soc.3(1) (1990), pp. 147–158

work page 1990
[5]

M.: Finding identities with the WZ method, J

Gessel I. M.: Finding identities with the WZ method, J. Symb. Comput.20(5) (1995), pp. 537–566

work page 1995
[6]

Feng L.-Q and Hou Q.-H.: Finding congruences with WZ method, Ramanujan J.68(3) (2025), pp. 64

work page 2025
[7]

Feng L.-Q and Hou Q.-H.: Some supercongruences of hyperheometric sums and the WZ method, J. Differ. Equ. Appl.31(2025), pp. 1–26

work page 2025
[8]

Guo V. J. W. and Liu J.-C.: Some congruences related to a congruence of Van Hamme, Integra Transforms and Special Functions,31(3) (2020), pp. 221-231

work page 2020
[9]

Mao G.-S.: Proof of some supercongruences via the Wilf-Zeilberger method, J. Differ. Equ. Appl.26(2020), pp. 1494–1513

work page 2020
[10]

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

work page 2012
[11]

Number Theory129(2009), pp

Zudilin, W.: Ramanujan-type supercongruences, J. Number Theory129(2009), pp. 1848–1857

work page 2009
[12]

Real Acad

Wang C., Hu D.-W.: Proof some supercongruences concerning truncated hypergeometric series, Rev. Real Acad. Clenc. Exactas Fis. Nat.Ser. A-Mat.,117(3) (2023), 99

work page 2023
[13]

and Zudilin W., ‘Divergent’ Ramanujan-type supercongruences, Proc

Guillera J. and Zudilin W., ‘Divergent’ Ramanujan-type supercongruences, Proc. Am. Math. Soc.140(3) (2012), pp. 765–777. 21

work page 2012
[14]

N., Generalized Hypergeometric Series, Cambridge University Press, Cambridge, (1935)

Bailey W. N., Generalized Hypergeometric Series, Cambridge University Press, Cambridge, (1935)

work page 1935
[15]

Wolstenholme J.: On certain properties of prime numbers. Q. J. Appl. Math.5(1862), pp. 35–39

work page
[16]

Mao G.-S.: Congruences involving Franel numbers and Ap´ ery-like numbers, temporarily on Researchgate (2023), preprint

work page 2023
[17]

Sun Z.-H.: Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete. Appl. Math.105(2000), pp. 193–223

work page 2000
[18]

Mao G.-S.: On some super-congruences for the coefficients of analytic solutions of certain differential equations, Indian J Pure Appl Math.57(2026), pp. 409–418

work page 2026
[19]

Math.9(1895), pp

Morely E.: Note on the congruence 2 4n ≡(−1) n(2n)!/(n!)2, where 2n+ 1 is a prime, Ann. Math.9(1895), pp. 168–170

work page
[20]

Internat

Sun Z.-W: A new series forπ 3 and related congruences. Internat. J. Math.26(8) (2015), 1550055 (23 pages)

work page 2015
[21]

Math.45, (1994), pp

Chu W.-C.: Inversion techniques and combinatorial identities: A uniffed treatment for the 7F6-series identities, Collect. Math.45, (1994), pp. 13–43

work page 1994
[22]

Integers: Electronic Journal of Combi- natorial Number Theory4(A22) (2004), pp

Granville A.: The square of the Fermat quotient. Integers: Electronic Journal of Combi- natorial Number Theory4(A22) (2004), pp. 1–3. 22

work page 2004

[1] [1]

Ramanujan S.: Modular equations and approximation toπ, Quart. J. Math.45(1914),pp. 350–372

work page 1914

[2] [2]

Guillera J.: A method for proving Ramanujan’s series for 1/π, Ramanujan J.52(2020), pp. 421–431

work page 2020

[3] [3]

Math.192, Dekker, New York, (1997), pp

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

work page 1996

[4] [4]

Wilf H. S. and Zeilberger D.: Rational functions certify combinatorial identities, J. Am. Math.Soc.3(1) (1990), pp. 147–158

work page 1990

[5] [5]

M.: Finding identities with the WZ method, J

Gessel I. M.: Finding identities with the WZ method, J. Symb. Comput.20(5) (1995), pp. 537–566

work page 1995

[6] [6]

Feng L.-Q and Hou Q.-H.: Finding congruences with WZ method, Ramanujan J.68(3) (2025), pp. 64

work page 2025

[7] [7]

Feng L.-Q and Hou Q.-H.: Some supercongruences of hyperheometric sums and the WZ method, J. Differ. Equ. Appl.31(2025), pp. 1–26

work page 2025

[8] [8]

Guo V. J. W. and Liu J.-C.: Some congruences related to a congruence of Van Hamme, Integra Transforms and Special Functions,31(3) (2020), pp. 221-231

work page 2020

[9] [9]

Mao G.-S.: Proof of some supercongruences via the Wilf-Zeilberger method, J. Differ. Equ. Appl.26(2020), pp. 1494–1513

work page 2020

[10] [10]

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

work page 2012

[11] [11]

Number Theory129(2009), pp

Zudilin, W.: Ramanujan-type supercongruences, J. Number Theory129(2009), pp. 1848–1857

work page 2009

[12] [12]

Real Acad

Wang C., Hu D.-W.: Proof some supercongruences concerning truncated hypergeometric series, Rev. Real Acad. Clenc. Exactas Fis. Nat.Ser. A-Mat.,117(3) (2023), 99

work page 2023

[13] [13]

and Zudilin W., ‘Divergent’ Ramanujan-type supercongruences, Proc

Guillera J. and Zudilin W., ‘Divergent’ Ramanujan-type supercongruences, Proc. Am. Math. Soc.140(3) (2012), pp. 765–777. 21

work page 2012

[14] [14]

N., Generalized Hypergeometric Series, Cambridge University Press, Cambridge, (1935)

Bailey W. N., Generalized Hypergeometric Series, Cambridge University Press, Cambridge, (1935)

work page 1935

[15] [15]

Wolstenholme J.: On certain properties of prime numbers. Q. J. Appl. Math.5(1862), pp. 35–39

work page

[16] [16]

Mao G.-S.: Congruences involving Franel numbers and Ap´ ery-like numbers, temporarily on Researchgate (2023), preprint

work page 2023

[17] [17]

Sun Z.-H.: Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete. Appl. Math.105(2000), pp. 193–223

work page 2000

[18] [18]

Mao G.-S.: On some super-congruences for the coefficients of analytic solutions of certain differential equations, Indian J Pure Appl Math.57(2026), pp. 409–418

work page 2026

[19] [19]

Math.9(1895), pp

Morely E.: Note on the congruence 2 4n ≡(−1) n(2n)!/(n!)2, where 2n+ 1 is a prime, Ann. Math.9(1895), pp. 168–170

work page

[20] [20]

Internat

Sun Z.-W: A new series forπ 3 and related congruences. Internat. J. Math.26(8) (2015), 1550055 (23 pages)

work page 2015

[21] [21]

Math.45, (1994), pp

Chu W.-C.: Inversion techniques and combinatorial identities: A uniffed treatment for the 7F6-series identities, Collect. Math.45, (1994), pp. 13–43

work page 1994

[22] [22]

Integers: Electronic Journal of Combi- natorial Number Theory4(A22) (2004), pp

Granville A.: The square of the Fermat quotient. Integers: Electronic Journal of Combi- natorial Number Theory4(A22) (2004), pp. 1–3. 22

work page 2004