Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher

Chen Wang; Victor J. W. Guo

REVIEW 2 major objections 3 minor 1 cited by

Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proves that the (E.2) and (F.2) supercongruences, along with two companion congruences, hold modulo $p^4$, with the extra term governed by Euler polynomials, and derives all four from one parameter-uniform $p$-adic identity.

desk verdict Genuine p^4 refinements of two Van Hamme and two Swisher supercongruences via a clean WZ argument; the proof is sound, only presentation gaps. read the letter →

arxiv 2501.09626 v1 pith:W34MIO2L submitted 2025-01-16 math.NT math.CO

Victor J. W. Guo , Chen Wang This is my paper · ORCID

classification math.NTmath.CO MSC 33C2011B7511B6533E50

keywords supercongruencetruncatedhypergeometricseriesEulerpolynomialsnumbersLegendresymbolharmonicq-congruencep-adiccongruences

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

A classical family of 13 hypergeometric supercongruences on truncated sums was known to hold modulo $p^3$; this paper upgrades four of them to modulus $p^4$. The main theorem states that for every prime $p>3$ and every $p$-adic integer $\alpha$, both the full sum up to $p-1$ and the truncated sum up to $\langle-\alpha\rangle_p$ are congruent modulo $p^4$ to $(-1)^a(\alpha+a)+(\alpha+a)^3E_{p-3}(\alpha)$, where $a=\langle-\alpha\rangle_p$ and $\alpha+a=pt$. Specializing $\alpha=1/3$ and $\alpha=1/4$ in the appropriate residue classes gives the four refined congruences. A conjectural $q$-analogue of one derived equality is also proposed.

What carries the argument

The load-bearing object is a pair of rational hypergeometric terms $F(n,k)$ and $G(n,k)$ satisfying the telescoping identity $F(n,k-1)-F(n,k)=G(n+1,k)-G(n,k)$. Summing this identity over $n$ and $k$ collapses the original hypergeometric sum into a small number of boundary terms, displayed as equation (3.2). Those boundary terms are evaluated by lemmas that compare products of Pochhammer symbols with harmonic numbers, and the final $p^3$ coefficient is identified with the Euler polynomial $E_{p-3}(\alpha)$, defined by the generating function $2e^{xt}/(e^t+1)$, through the congruence $\sum_{k=1}^{a}(-1)^k k^{p-3}\equiv\frac{(-1)^a}{2}E_{p-3}(\alpha)\pmod p$.

What would settle it

Take $p=11$ and $\alpha=1/4$, so $a=8$; compute both sides of (1.12) modulo $11^4$ by evaluating the finite rational sum and $E_8(1/4)$ exactly (all denominators are integers below 11, hence invertible modulo $11^4$). A mismatch in the $p^3$ coefficient would refute the theorem.

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Extended reading notes

Core claim

The central discovery is a single parameter-uniform congruence (Theorem 1.5). For any prime $p>3$ and any $p$-adic integer $\alpha$, writing $a=\langle-\alpha\rangle_p$ and $\alpha+a=pt$, the paper proves $$\sum_{k=0}^{M}(-1)^k(2k+\$\alpha$)\frac{(\$\alpha$)$_k^{3}$}{(1)$_k^{3}$}\equiv(-1)^a(\$\alpha$+a)+(\$\alpha$+a)^3E_{p-3}(\$\alpha$)\pmod{$p^{4}$},$$ where $M$ is either $p-1$ or $a$. The proof reduces the sum to boundary terms through a telescoping identity for a pair of rational hypergeometric terms, then evaluates those boundary terms using congruences for harmonic numbers and identities for Euler polynomials. The cases $\alpha=1/3$ and $\alpha=1/4$ in the four residue classes recover the refinements of the (E.2), (F.2) and companion supercongruences.

Load-bearing premise

The proof turns on the exact correctness of the unproved rational-function identity (3.1) and of the four harmonic-binomial identities in Lemma 2.5; if any of these has a hidden exception, the $p^4$ cancellation fails.

Editorial extensions

If this is right

The (E.2) and (F.2) supercongruences hold modulo $p^4$, with explicit corrections $\frac{p^3}{9}E_{p-3}(1/3)$ and $\frac{p^3}{16}E_{p-3}(1/4)$.
The two companion congruences also hold modulo $p^4$, with corrections $\frac{8p^3}{9}E_{p-3}(1/3)$ and $\frac{27p^3}{16}E_{p-3}(1/4)$ in the remaining residue classes.
For any admissible $\alpha$, the full sum and the truncated sum are congruent modulo $p^4$, so the tail of the series from $k=a+1$ to $p-1$ vanishes modulo $p^4$.
Combining one refined congruence with another known $p^4$ congruence yields an equality modulo $p^4$ between two different truncated hypergeometric sums (equation (4.2)).
A conjectural $q$-analogue of that equality is stated modulo $[n]\Phi_n(q)^3$ for $n\equiv1\pmod4$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The parameter-uniform shape of the theorem suggests that the same telescoping strategy may yield $p^4$ refinements of other entries in the classical supercongruence list by choosing $\alpha$ to match other rational parameters; the paper does not pursue these cases.
The conjecture that the $q$-analogue holds modulo $[n]\Phi_n(q)^3$ while only the modulus $[n]\Phi_n(q)^2$ cases are known points to a gap that the proof technique of the main theorem might help close.
A testable extension is to compute numerical values beyond modulus $p^4$ to see whether the coefficient of $p^5$ in a further refinement follows a similar Euler-polynomial pattern, which the paper's method does not address.

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Formalized claims in Lean

Claim #1: The central discovery is a single parameter-uniform congruence (Theorem 1.5). For any prime $p>3$ and any $p$-adic integer $\alpha$, writing $a=\langle-\alpha\rangle_p$ and $\alpha+a=pt$, the paper proves $$\sum_{k=0}^{M}(-1)^k(2k+\$\alpha$)\frac{(\$\alpha$)$_k^{3}$}{(1)$_k^{3}$}\equiv(-1)^a(\$\alpha$+a)+(\$\alpha$+a)^3E_{p-3}(\$\alpha$)\pmod{$p^{4}$},$$ where $M$ is either $p-1$ or $a$. The proof

/-- @claim 1 The central discovery is a single parameter-uniform congruence (Theorem 1.5). For any prime $p>3$ and any $p$-adic integer $\alpha$, writing $a=\langle-\alpha\rangle_p$ and $\alpha+a=pt$, the paper proves $$\sum_{k=0}^{M}(-1)^k(2k+\$\alpha$)\frac{(\$\alpha$)$_k^{3}$}{(1)$_k^{3}$}\equiv(-1)^a(\$\alpha$+a)+(\$\alpha$+a)^3E_{p-3}(\$\alpha$)\pmod{$p^{4}$},$$ where $M$ is either $p-1$ or $a$. The proof -/
def central_claim : Prop :=

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: Theorem 1.5 is the real result here — a uniform modulus-p^4 congruence for arbitrary p-adic α, with Euler-polynomial correction. Theorems 1.1–1.4 are honest corollaries, and the paper correctly notes prior p^4 work only covered special cases. The WZ setup is natural; the cancellations in Section 3 check out, and the reduction to boundary terms via (3.2) is the right engine. This is a solid advance in the specialized supercongruence literature, not a revolution.

What it does well: it generalizes Sun's p^4 refinement of (B.2) to (E.2), (F.2), and two Swisher congruences; Theorem 1.5 is new for arbitrary α; the proof treats the three residue classes cleanly; the Euler-polynomial bookkeeping at the end is neat. Self-citations are contextual and not load-bearing. The paper is carefully organized and the p-adic expansions look right.

Soft spots, in proportion: the WZ identity (3.1) is asserted as 'easily verified' without displaying the rational-function certificate. For a referee that is a minor annoyance, not a flaw: given the explicit F and G, (3.1) is a routine rational-function equality; any CAS or hand check confirms it. Likewise Lemma 2.5 lists four harmonic-binomial identities without proofs, citing known results and Sigma. They are standard and can be verified independently; the paper even points to exact sources for two of them. The stress-test note worries these are load-bearing. They carry part of the p^4 cancellation, but they are elementary finite identities, not hidden conjectures.

The only mild weakness I see is a presentation one: the paper could display the WZ certificate in an appendix or state Lemma 2.5 with proof sketches, and it should define convergence or p-adic interpretation more carefully when discussing Euler polynomial values at p-adic α. None of this threatens the main theorem; I did not find a load-bearing error.

Bottom line: for researchers in p-adic hypergeometric congruences, this is worth reading and citing. The q-congruence conjecture is a reasonable open problem. It deserves a serious referee; I would accept it after minor revision.

Referee Report

2 major / 3 minor

Summary. This paper proves refinements modulo p^4 of four known supercongruences for truncated hypergeometric series: Van Hamme's (E.2) and (F.2) and two later supercongruences by Swisher. The main result, Theorem 1.5, states a unified congruence for any p-adic integer α and primes p>3: both the full sum and the truncated sum over k=0,...,⟨−α⟩_p equal (−1)^a(α+a)+(α+a)^3 E_{p−3}(α) modulo p^4, where a=⟨−α⟩_p. The proof uses a WZ pair to reduce the sums to boundary terms, then applies p-adic expansions involving harmonic numbers and binomial-harmonic identities to extract the p^4 correction. Theorems 1.1–1.4 are specializations α=1/3,1/4. The paper closes with a q-congruence conjecture.

Significance. If the missing verifications are supplied, the paper is a solid contribution: it gives a common p^4 refinement for four previously independent p^3 theorems, extends Sun's WZ approach beyond the (B.2) case, and sharpens the correction terms through Euler polynomials. The main theorem is explicit and falsifiable, and its specialization yields concrete congruences suitable for numerical checking. The q-conjecture in Section 4 is a natural next step and is supported by the known mod [n]Φ_n(q)^2 results.

major comments (2)

[Section 3, Eq. (3.1)] The WZ identity F(n,k−1)−F(n,k)=G(n+1,k)−G(n,k) is asserted as 'easily verified' without displaying the rational-function certificate. Because this identity is the sole input that converts the hypergeometric sum into boundary terms in (3.2), please provide the certificate or a complete verification, and spell out the conventions for (1)_m with m<0 in the definition of G.
[Lemma 2.5] The four harmonic-binomial identities (2.1)–(2.4) are quoted as known or Sigma-checkable but are not proved. They are used essentially in Lemma 2.6 and Lemma 2.8 to obtain the cancellations that produce the p^4 term. Please supply proofs or exact references with theorem/equation numbers for each identity, rather than a general statement that they are automatic.

minor comments (3)

[Section 3, proof of (1.13)] In the displayed tail-summation equality, the factor (1+p(t+1))_k should be (1+pt)_k; the subsequent argument is unaffected because only the value modulo p is used.
[Lemma 2.6] The final congruence in the proof is labeled (mod p^2), whereas the lemma states (mod p^4); the intended meaning is that the bracketed sum is determined modulo p^2 and the factor p^3t^3 upgrades the modulus.
[Lemma 2.2] The identity E_{2n}(0)=E_{2n}(1)=0 should specify n≥1, since E_0(0)=1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the proof is a self-contained WZ derivation whose inputs are independent lemmas, not the target congruence.

full rationale

Walk through the chain: Theorem 1.5 is proved by a WZ pair (F,G), identity (3.1), summation to (3.2), then Lemmas 2.3–2.8. These lemmas rely on Lehmer harmonic congruences (Lemma 2.1), Euler polynomial identities (Lemma 2.2), and harmonic-binomial identities (Lemma 2.5) attributed to Gould, Dilcher, Van Hamme, or checkable by Sigma; none of these are the target theorem or fitted to it. Self-references [6,8,9,24] in the introduction and Section 4 are contextual remarks about prior p^3 and q-versions and are not invoked in the proof of Theorem 1.5. The final appearance of E_{p-3}(α) comes from Lemma 2.2's evaluation of sum (-1)^k k^{p-3}, an independent Euler-polynomial identity, not from defining E_{p-3} via the congruence. The only unverified assertion is the rational-function WZ identity (3.1), stated as 'easily verified'; even if one wanted a certificate, an unverified algebraic identity is a correctness risk, not circularity, because it is not assumed to be the target result. No fitted input is renamed as a prediction, and no uniqueness theorem is imported. Hence score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical or mathematical entities. The central proof rests on standard harmonic-number and Euler-polynomial results, plus two computational identities, the WZ identity and the Lemma 2.5 identities, that are asserted rather than fully derived in the text.

assumptions (4)

standard math Lehmer's harmonic-number congruences: H_{p-1} ≡ 0 (mod p^2) and H_{p-1}^{(2)} ≡ H_{(p-1)/2}^{(2)} ≡ 0 (mod p).
Invoked in Lemma 2.3, Lemma 2.4, Lemma 2.7, and Lemma 2.8 to discard low-order p-adic terms.
standard math Euler polynomial identities: E_{2n}(0)=E_{2n}(1)=0, E_n(1-x)=(-1)^n E_n(x), and the finite-sum identity for ∑ (-1)^k k^m.
Used in Lemma 2.2 and again in the final step of the proof of Theorem 1.5 to convert a harmonic sum into E_{p-3}(α).
standard math The four harmonic-binomial identities in Lemma 2.5, including ∑_{k=1}^n (-1)^k/(k^2 C(n,k)) = H_n^{(2)} + 2∑(-1)^k/k^2.
The authors state these 'should be known' and checkable by Sigma, but no proof is given in the paper. They are used in Lemmas 2.6 and 2.8 during the p^4 cancellation.
ad hoc to paper The WZ telescoping identity F(n,k-1)-F(n,k) = G(n+1,k)-G(n,k) for the defined F and G.
Equation (3.1) is the backbone of the proof; the authors say it can be easily verified but do not display the rational-function certificate. If this identity failed, the boundary-sum reduction in (3.2) would not hold.

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Cite this review

Pith. "Pith review of Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher." pith.science (2026). https://pith.science/paper/W34MIO2L

@misc{pith2026250109626,
  author       = {Pith},
  title        = {Pith review of: Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W34MIO2L}},
  note         = {Machine review of arXiv:2501.09626}
}

abstract

In 1997, Van Hamme proposed 13 supercongruences on truncated hypergeometric series. Van Hamme's (B.2) supercongruence was first confirmed by Mortenson and received a WZ proof by Zudilin later. In 2012, using the WZ method again, Sun extended Van Hamme's (B.2) supercongruence to the modulus $p^4$ case, where $p$ is an odd prime. In this paper, by using a more general WZ pair, we generalize Hamme's (E.2) and (F.2) supercongruences, as well as two supercongruences by Swisher, to the modulus $p^4$ case. Our generalizations of these supercongruences are related to Euler polynomials. We also put forward a relevant conjecture on $q$-congruences for further study.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages · cited by 1 Pith paper

[1]

Bauer, Von den Coeﬃcienten der Reihen von Kugelfuncti onen einer Variablen, J

G. Bauer, Von den Coeﬃcienten der Reihen von Kugelfuncti onen einer Variablen, J. Reine Angew. Math. 56 (1859), 101–121. 13

work page
[2]

Borwein and P.B

J.M. Borwein and P.B. Borwein, Pi and the AGM, a study in an alytic number theory and computational complexity, reprint of the 1987 original, Ca nadian Mathematical Society Series of Monographs and Advanced Texts, volume 4, John Wile y & Sons, Inc., New York, 1998

work page 1987
[3]

Chen, X.Y

Y.G. Chen, X.Y. Xie, and B. He, On some congruences of cert ain binomial sums, Ramanu- jan J. (2016) 40, 237–244

work page 2016
[4]

Dilcher, Some q-series identities related to divisor functions, Discrete Math

K. Dilcher, Some q-series identities related to divisor functions, Discrete Math. 145 (1995), 83–93

work page 1995
[5]

Gould, Combinatorial Identities, Morgantown Prin ting and Binding Co., 1972

H.W. Gould, Combinatorial Identities, Morgantown Prin ting and Binding Co., 1972

work page 1972
[6]

Guo, q-Analogues of the (E.2) and (F.2) supercongruences of Van Ha mme, Ra- manujan J

V.J.W. Guo, q-Analogues of the (E.2) and (F.2) supercongruences of Van Ha mme, Ra- manujan J. 49 (2019), 531–544

work page 2019
[7]

Guo, q-Supercongruences modulo the fourth power of a cyclotomic p olynomial via creative microscoping, Adv

V.J.W. Guo, q-Supercongruences modulo the fourth power of a cyclotomic p olynomial via creative microscoping, Adv. Appl. Math. 120 (2020), Art. 10 2078

work page 2020
[8]

Guo and J.-C

V.J.W. Guo and J.-C. Liu, Some congruences related to a co ngruence of Van Hamme, Integral Transforms Spec. Funct. 31 (2020), 221–231

work page 2020

Show all 27 references

[9]

Guo and W

V.J.W. Guo and W. Zudilin, A q-microscope for supercongruences, Adv. Math. 346 (2019), 329–358

2019
[10]

He, Some congruences on truncated hypergeometric se ries, Proc

B. He, Some congruences on truncated hypergeometric se ries, Proc. Amer. Math. Soc. 143 (2015), 5173–5180

2015
[11]

Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann

E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350–360

1938
[12]

Long, Hypergeometric evaluation identities and sup ercongruences, Paciﬁc J

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

2011
[13]

Magnus, F

W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Func- tions of Mathematical Physics (3rd edition), Springer, New York, 1966

1966
[14]

Mao, Proof of two supercongruences of truncated hy pergeometric series 4F3, Acta Math

G.S. Mao, Proof of two supercongruences of truncated hy pergeometric series 4F3, Acta Math. Sin. (Engl. Ser.) 40 (2024), 1015–1028

2024
[15]

Mortenson, A p-adic supercongruence conjecture of van Hamme, Proc

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

2008
[16]

Ramanujan, Modular equations and approximation to π , Quart

S. Ramanujan, Modular equations and approximation to π , Quart. J. Math. 45 (1914), 350–372

1914
[17]

Schneider, Symbolic summation assists combinatori cs, S´ em

C. Schneider, Symbolic summation assists combinatori cs, S´ em. Lothar. Combin. 56 (2007), B56b

2007
[18]

Sun, Super congruences and Euler numbers, Sci

Z.-W. Sun, Super congruences and Euler numbers, Sci. Ch ina Math. 54 (2011), 2509–2535

2011
[19]

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

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

2012
[20]

Sun, Open conjectures on congruences

Z.-W. Sun, Open conjectures on congruences. J. Nanjing Univ. Math. Biquart. 36 (2019), 1–99

2019
[21]

Swisher, On the supercongruence conjectures of van H amme, Res

H. Swisher, On the supercongruence conjectures of van H amme, Res. Math. Sci. 2 (2015), Art. 18

2015
[22]

Van Hamme, Advanced problem 6407, Amer

L. Van Hamme, Advanced problem 6407, Amer. Math. Monthl y 40 (1982), 703–704. 14

1982
[23]

Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in: p-Adic functional analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl

L. Van Hamme, 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

1996
[24]

Wang and Z.-W

C. Wang and Z.-W. Sun, p-Adic analogues of hypergeometric identities and their ap- plications, Nanjing Univ. J. Math. Biquarterly 41 (2024), n o. 1, 34–56. (See also arXiv:1910.06856)

2024 arXiv
[25]

Wilf and D

H.S. Wilf and D. Zeilberger, An algorithmic proof theor y for hypergeometric (ordinary and “q”) multisum/integral identities, Invent. Math. 108 (1992) , 575–633

1992
[26]

Wilf and D

H.S. Wilf and D. Zeilberger, Rational function certiﬁc ation of multisum/integral/“ q” iden- tities, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 148–153

1992
[27]

Zudilin, Ramanujan-type supercongruences, J

W. Zudilin, Ramanujan-type supercongruences, J. Numb er Theory 129 (2009), 1848–1857. 15

2009

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Bauer, Von den Coeﬃcienten der Reihen von Kugelfuncti onen einer Variablen, J

G. Bauer, Von den Coeﬃcienten der Reihen von Kugelfuncti onen einer Variablen, J. Reine Angew. Math. 56 (1859), 101–121. 13

work page

[2] [2]

Borwein and P.B

J.M. Borwein and P.B. Borwein, Pi and the AGM, a study in an alytic number theory and computational complexity, reprint of the 1987 original, Ca nadian Mathematical Society Series of Monographs and Advanced Texts, volume 4, John Wile y & Sons, Inc., New York, 1998

work page 1987

[3] [3]

Chen, X.Y

Y.G. Chen, X.Y. Xie, and B. He, On some congruences of cert ain binomial sums, Ramanu- jan J. (2016) 40, 237–244

work page 2016

[4] [4]

Dilcher, Some q-series identities related to divisor functions, Discrete Math

K. Dilcher, Some q-series identities related to divisor functions, Discrete Math. 145 (1995), 83–93

work page 1995

[5] [5]

Gould, Combinatorial Identities, Morgantown Prin ting and Binding Co., 1972

H.W. Gould, Combinatorial Identities, Morgantown Prin ting and Binding Co., 1972

work page 1972

[6] [6]

Guo, q-Analogues of the (E.2) and (F.2) supercongruences of Van Ha mme, Ra- manujan J

V.J.W. Guo, q-Analogues of the (E.2) and (F.2) supercongruences of Van Ha mme, Ra- manujan J. 49 (2019), 531–544

work page 2019

[7] [7]

Guo, q-Supercongruences modulo the fourth power of a cyclotomic p olynomial via creative microscoping, Adv

V.J.W. Guo, q-Supercongruences modulo the fourth power of a cyclotomic p olynomial via creative microscoping, Adv. Appl. Math. 120 (2020), Art. 10 2078

work page 2020

[8] [8]

Guo and J.-C

V.J.W. Guo and J.-C. Liu, Some congruences related to a co ngruence of Van Hamme, Integral Transforms Spec. Funct. 31 (2020), 221–231

work page 2020