---
title: Van Hamme Supercongruences
url: https://www.emergentmind.com/topics/van-hamme-supercongruences
type: topic
---

# Van Hamme Supercongruences

Searching arXiv for recent and foundational papers on Van Hamme supercongruences to ground the article.
Van Hamme supercongruences are \(p\)-adic analogues of Ramanujan-type hypergeometric identities, formulated in 1997 as thirteen conjectures labeled \((\mathrm{A}.2)\)–\((\mathrm{M}.2)\). Their basic form is a congruence between a truncated hypergeometric or Ramanujan-type sum and a simple \(p\)-adic expression—often involving the Morita \(p\)-adic gamma function \(\Gamma_p\)—modulo a surprisingly high power of a prime. In current usage, they are understood as lying at the intersection of Ramanujan-type formulas for \(1/\pi\), truncated hypergeometric series, \(p\)-adic analysis, modular forms, and motives [1504.01976][2201.04297][2408.02002].

## 1. Origin and defining structure

The starting point is Ramanujan’s hypergeometric theory of series for \(1/\pi\). A prototypical analytic identity is
\[
\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},
\]
with \((a)_n=\Gamma(a+n)/\Gamma(a)\) the Pochhammer symbol [1504.01976]. Van Hamme’s insight was that such series admit \(p\)-adic shadows: if one truncates at a prime-dependent bound such as \(p-1\), \((p-1)/2\), \((p-1)/3\), or \((p-1)/4\), then the resulting rational number frequently satisfies a congruence modulo \(p^k\) with \(k\) much larger than naive valuation estimates suggest.

In this setting, a supercongruence is a statement of the form
\[
S_p \equiv \text{(simple \(p\)-adic expression)} \pmod{p^r},
\]
where \(S_p\) is a truncated hypergeometric sum and the right-hand side is often expressed through \(\Gamma_p\) [2006.16929]. Van Hamme’s original list organizes these statements as \(p\)-adic analogues of Ramanujan-type identities. The labels \((\mathrm{A}.2)\)–\((\mathrm{M}.2)\) encode membership in that list, while “.2” refers to the Ramanujan-type \(1/\pi\) formulas of “second kind” in Van Hamme’s classification for entries such as \((\mathrm{G}.2)\) [2201.04297].

A standard structural feature is the coexistence of three layers. The analytic layer is an infinite hypergeometric identity; the arithmetic layer is a finite truncation; the \(p\)-adic layer is a congruence to a gamma-value expression or a simpler algebraic quantity. In several cases the right-hand side is not merely an integer multiple of \(p\), but a nontrivial \(\Gamma_p\)-combination reflecting the same parameters as the underlying hypergeometric series [1403.5232].

## 2. Representative examples and arithmetic patterns

Several entries have become paradigmatic because they illustrate the range of behaviors present in Van Hamme’s list.

| Label | Truncated sum | Congruence pattern |
|---|---|---|
| \((\mathrm{B}.2)\) | \(\sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\) | \(\equiv p(-1)^{(p-1)/2}\pmod{p^3}\) |
| \((\mathrm{H}.2)\) | \(\sum_{k=0}^{(p-1)/2}A_k\) | \(\equiv -\Gamma_p(1/4)^4\) or \(0\) mod \(p^2\) |
| \((\mathrm{G}.2)\) | \(\sum_{k=0}^{(p-1)/4}(8k+1)\frac{(1/4)_k^4}{k!^4}\) | \(\equiv p\,\Gamma_p(1/2)\Gamma_p(1/4)/\Gamma_p(3/4)\pmod{p^3}\) |
| \((\mathrm{K}.2)\) | \(\sum_{n=0}^{p-1}\frac{(1/2)_n^3}{n!^3}(42n+5)64^{-n}\) | \(\equiv 5p(-1)^{(p-1)/2}\pmod{p^4}\) |

Here
\[
A_k=\prod_{j=0}^{k-1}\left(\frac{\frac12+j}{1+j}\right)^3
=\frac{1}{2^{6k}\binom{2k}{k}^3},
\]
the standard hypergeometric kernel appearing in \((\mathrm{B}.2)\) and \((\mathrm{H}.2)\) [1910.10932].

The \((\mathrm{H}.2)\) example shows that Van Hamme supercongruences are not uniformly of “Ramanujan \(1/\pi\)-type” on the right-hand side. The same truncated sum can also be written as
\[
\sum_{k=0}^{(p-1)/2}A_k \equiv a(p)\pmod{p^2},
\]
where \(a(p)\) is the \(p\)-th Fourier coefficient of the weight-3 CM modular form
\[
f(q)=q\prod_{j=1}^{\infty}(1-q^{4j})^6,
\]
so the congruence admits both a \(p\)-adic gamma formulation and a modular-form formulation [1910.10932].

The \((\mathrm{G}.2)\) case is central because it sits exactly at the meeting point of Ramanujan-type formulas for \(1/\pi\), truncated \({}_4F_3\)-type series, and \(p\)-adic gamma phenomena. For \(p\equiv1\pmod4\),
\[
\sum_{k=0}^{(p-1)/4}(8k+1)\frac{(1/4)_k^4}{k!^4}
\equiv
p\,\frac{\Gamma_p(1/2)\Gamma_p(1/4)}{\Gamma_p(3/4)}
\pmod{p^3},
\]
and Swisher strengthened this to modulus \(p^4\) [2201.04297].

Residue classes mod \(4\) or mod \(6\) are frequently essential. A vanishing statement in one residue class is often only the first visible term of a deeper expansion. For example, Liu sharpened both \((\mathrm{A}.2)\) and \((\mathrm{H}.2)\) for \(p\equiv3\pmod4\) from lower-order vanishing congruences to explicit modulo-\(p^4\) formulas, revealing previously hidden \(p\)-adic structure [1807.03987]. This corrects the common impression that the “zero” cases are arithmetically degenerate.

## 3. Proof techniques and structural mechanisms

No single proof method governs the entire list. The literature instead exhibits a stable toolkit whose components interact in different proportions according to the label.

A first major line uses classical hypergeometric transformation formulae to rewrite terminating series as Gamma-quotients, then converts those to \(\Gamma_p\)-quotients and expands \(p\)-adically. Long and Ramakrishna made this strategy explicit: classical identities such as Pfaff, Kummer, Gauss, Whipple, and Dougall transform truncated hypergeometric expressions into ratios of classical \(\Gamma\)-values; a bridge lemma replaces them by \(\Gamma_p\)-quotients; and Taylor expansions of \(\Gamma_p\) expose cancellations of low-order terms, producing the “super” modulus [1403.5232]. Their expansion
\[
\frac{\Gamma_p(a+bp^r)}{\Gamma_p(a)}
\equiv
\sum_{k=0}^{t}\frac{G_k(a)}{k!}(-bp^r)^k
\pmod{p^{(t+1)r}}
\]
is the local analytic engine behind many refinements [1403.5232].

A second line uses the Wilf–Zeilberger method. In the proof of \((\mathrm{K}.2)\), Osburn and Zudilin used a WZ pair due to Guillera together with Wolstenholme’s congruence, Morley’s congruence, and harmonic-sum identities to prove the last remaining case of Van Hamme’s original thirteen conjectures [1504.01976]. More recently, a systematic “streamlined WZ” framework has been developed: suitably chosen WZ devices, followed by Long–Ramakrishna \(p\)-adic gamma approximations, furnish uniform proofs of \((\mathrm{B}.2)\), \((\mathrm{C}.2)\), \((\mathrm{D}.2)\), \((\mathrm{E}.2)\), \((\mathrm{F}.2)\), \((\mathrm{G}.2)\), and \((\mathrm{H}.2)\), while \((\mathrm{I}.2)\) becomes a special case where Gosper’s algorithm already succeeds [2508.00343].

A third line is explicitly combinatorial. Liu’s refinements of \((\mathrm{A}.2)\) and \((\mathrm{H}.2)\) for \(p\equiv3\pmod4\) were derived from hypergeometric identities combined with Sigma-discovered binomial-harmonic identities and \(p\)-adic gamma manipulations [1807.03987]. Guo and Wang’s refinements of \((\mathrm{E}.2)\), \((\mathrm{F}.2)\), and two Swisher congruences use a more general WZ pair and identify Euler polynomials \(E_{p-3}(a)\) as the universal correction term at the \(p^4\)-level [2501.09626].

Despite the success of these methods, the literature repeatedly emphasizes a conceptual gap: even after the last original case \((\mathrm{K}.2)\) was proved, there was still no known general framework explaining why Ramanujan-type series should systematically produce supercongruences of the observed strength [1504.01976].

## 4. \(q\)-analogues and cyclotomic deformation

A large modern branch of the subject studies \(q\)-supercongruences, where ordinary factorials are replaced by \(q\)-shifted factorials,
\[
(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1}),
\]
ordinary integers by \(q\)-integers
\[
[n]=\frac{1-q^n}{1-q},
\]
and powers of \(p\) by powers of cyclotomic polynomials \(\Phi_n(q)\) or by mixed moduli such as \([n]\Phi_n(q)^r\) [2006.16929]. The heuristic is that, when \(n=p\) is prime and \(q\to1\), congruences modulo \(\Phi_p(q)^r\) recover congruences modulo \(p^r\).

This deformation is not merely formal. It creates a setting in which congruence classes become polynomial ideals and hypergeometric transformations become basic hypergeometric transformations. Guo and Zudilin gave a common \(q\)-analogue whose specializations \(q=-1\) and \(q=1\) recover \((\mathrm{B}.2)\) and \((\mathrm{H}.2)\), showing that a Ramanujan \(1/\pi\)-type congruence and a modular-form-valued congruence can be encoded by one basic-hypergeometric identity [1910.10932].

For \((\mathrm{H}.2)\), Wei derived the missing modulus-\(\Phi_n(q)^3\) \(q\)-analogue in the class \(n\equiv1\pmod4\), complementing Guo’s earlier \(n\equiv3\pmod4\) result and thereby covering all odd \(n\) at the \(p^3\)-level [2006.16929]. For \((\mathrm{J}.2)\), creative microscoping produced a complete \(q\)-analogue modulo the fourth power of a cyclotomic polynomial, confirming a conjecture of Guo [1912.00765].

The \((\mathrm{G}.2)\) line is especially developed on the \(q\)-side. Liu and Wang constructed two \(q\)-analogues of Swisher’s \(p^4\) strengthening and a master parametric \(q\)-congruence modulo \([n]\Phi_n(q)^3\); their formulas recover Swisher’s congruence under \(n=p\) and \(q\to1\) [2201.04297]. A further 2026 synthesis via the \(q\)-Zeilberger algorithm unified the \(q\)-analogues of \((\mathrm{C}.2)\) and \((\mathrm{G}.2)\), lifted the modulus from \([n]\Phi_n(q)^3\) to \([n]\Phi_n(q)^4\), and extracted \(p^5\)-level refinements involving Bernoulli numbers in the classical limit [2603.26223].

The main technical motifs on the \(q\)-side are Watson’s \({}_8\phi_7\) transformation, Andrews’s and Jain’s \(q\)-Whipple formulas, the \(q\)-Dixon sum, Jackson’s \({}_6\phi_5\), polynomial Chinese remainder theorems for coprime factors such as \((1-aq^n)(a-q^n)\), and the creative microscoping paradigm introduced by Guo and Zudilin [2201.04297][2006.16929][1912.00765].

## 5. Refinements, parametric families, and multidimensional extensions

A conspicuous recent trend is the passage from isolated congruences to parametric and multidimensional families. The parametric viewpoint often reveals that a classical Van Hamme congruence is one specialization of a larger identity with deformation parameters, and that higher-order corrections are governed by familiar arithmetic objects.

Guo and Wang generalized \((\mathrm{E}.2)\), \((\mathrm{F}.2)\), and two Swisher supercongruences from modulus \(p^3\) to modulus \(p^4\), with correction terms expressed through Euler polynomials \(E_{p-3}(a)\) [2501.09626]. On the \(q\)-side, a refined unified \(q\)-analogue of \((\mathrm{B}.2)\), \((\mathrm{E}.2)\), and \((\mathrm{F}.2)\) was then obtained modulo \([n]\Phi_n(q)^3\), and its \(q\to1\) specialization recovers the corresponding parametric Euler-polynomial supercongruence [2505.04395].

Another direction replaces single sums by double or triple basic hypergeometric series. For kernels associated with \((\mathrm{D}.2)\), Wei proved double- and triple-series \(q\)-supercongruences modulo the sixth power of a cyclotomic polynomial, and the specialization \(q\to1\), \(n=p\equiv1\pmod6\) yielded classical double and triple supercongruences modulo \(p^6\) [2408.02002]. A related 2024 paper established further \(q\)-supercongruences for multiple basic hypergeometric series modulo the fifth and sixth powers of cyclotomic polynomials, including double-sum generalizations attached to \((\mathrm{C}.2)\), Long’s supercongruence, and double/triple conclusions associated with \((\mathrm{D}.2)\) [2408.07226].

These developments suggest that Van Hamme’s program is no longer confined to one-dimensional truncations. Multiple convolutions of the same basic kernel can still satisfy high-power congruences, and the same machinery—creative microscoping, polynomial CRT, and terminating transformation formulas—continues to operate. A plausible implication is that the relevant arithmetic structure is attached more to the hypergeometric kernel itself than to the one-dimensional truncation alone.

## 6. Broader landscape, extensions, and unresolved structure

Historically, the original thirteen conjectures were all proved by 2015, with \((\mathrm{K}.2)\) the final case [1504.01976]. Yet the subject did not stabilize into a closed theory; instead it expanded in several directions.

One direction concerns unification. The 2025 streamlined WZ framework indicates that a substantial portion of the list can be handled by a common proof architecture rather than by isolated ad hoc arguments [2508.00343]. Another concerns arithmetic refinements: \((\mathrm{G}.2)\) has been pushed from modulus \(p^3\) to \(p^4\), and then to \(p^5\)-level expansions in the \(q\)-to-\(1\) limit; \((\mathrm{H}.2)\) has modulus-\(p^3\) refinements; \((\mathrm{E}.2)\) and \((\mathrm{F}.2)\) now admit \(p^4\)-level Euler-polynomial corrections [2201.04297][1807.03987][2501.09626].

A further extension is geometric and algebraic. Guillera’s “mosaic supercongruences” generalize Van Hamme–Zudilin patterns to Ramanujan-Sato-type series involving simple square roots anywhere in the summand, with the truncated sums decomposing into components in multiquadratic fields and each component satisfying its own Van Hamme-style congruence. These examples are numerical and conjectural rather than proved, but they explicitly suggest that the classical one-component pattern may be only a special case of a broader multi-component phenomenon [1007.2290].

The central unresolved issue remains explanatory rather than computational. The papers repeatedly note the abundance of examples and the success of several powerful techniques, but also the absence of a general conceptual framework that predicts the exact modulus or explains uniformly why Ramanujan-type periods, modular forms, \(p\)-adic gamma values, and truncated hypergeometric sums align so consistently. The current state of the field therefore combines a largely completed foundational list with an active and technically sophisticated research program on refinements, \(q\)-deformations, parametric families, and higher-dimensional extensions [1504.01976][1403.5232].

Source: https://www.emergentmind.com/topics/van-hamme-supercongruences