Van Hamme Supercongruences

• Van Hamme supercongruences are p-adic analogues of Ramanujan-type identities that reveal deep connections between truncated hypergeometric series, modular forms, and p-adic gamma functions.
• The subject employs classical hypergeometric transformations, Wilf–Zeilberger methods, and combinatorial techniques to establish congruences modulo high powers of primes, often incorporating q-analogues and cyclotomic deformations.
• Recent advancements extend the theory to parametric families and multidimensional series, yielding refined congruences and uncovering intricate arithmetic patterns that challenge existing theoretical frameworks.

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 (Osburn et al., 2015, Liu et al., 2022, Wei, 2024).

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 $(\mathrm{A}.2)$0 the Pochhammer symbol (Osburn et al., 2015). Van Hamme’s insight was that such series admit $(\mathrm{A}.2)$1-adic shadows: if one truncates at a prime-dependent bound such as $(\mathrm{A}.2)$2, $(\mathrm{A}.2)$3, $(\mathrm{A}.2)$4, or $(\mathrm{A}.2)$5, then the resulting rational number frequently satisfies a congruence modulo $(\mathrm{A}.2)$6 with $(\mathrm{A}.2)$7 much larger than naive valuation estimates suggest.

In this setting, a supercongruence is a statement of the form

$(\mathrm{A}.2)$8

where $(\mathrm{A}.2)$9 is a truncated hypergeometric sum and the right-hand side is often expressed through $(\mathrm{M}.2)$0 (Wei, 2020). Van Hamme’s original list organizes these statements as $(\mathrm{M}.2)$1-adic analogues of Ramanujan-type identities. The labels $(\mathrm{M}.2)$2–$(\mathrm{M}.2)$3 encode membership in that list, while “.2” refers to the Ramanujan-type $(\mathrm{M}.2)$4 formulas of “second kind” in Van Hamme’s classification for entries such as $(\mathrm{M}.2)$5 (Liu et al., 2022).

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 $(\mathrm{M}.2)$6-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 $(\mathrm{M}.2)$7, but a nontrivial $(\mathrm{M}.2)$8-combination reflecting the same parameters as the underlying hypergeometric series (Long et al., 2014).

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{M}.2)$9	$p$0	$p$1
$p$2	$p$3	$p$4 or $p$5 mod $p$6
$p$7	$p$8	$p$9
$p$0	$p$1	$p$2

Here

$p$3

the standard hypergeometric kernel appearing in $p$4 and $p$5 (Guo et al., 2019).

The $p$6 example shows that Van Hamme supercongruences are not uniformly of “Ramanujan $p$7-type” on the right-hand side. The same truncated sum can also be written as

$p$8

where $p$9 is the $\Gamma_p$0-th Fourier coefficient of the weight-3 CM modular form

$\Gamma_p$1

so the congruence admits both a $\Gamma_p$2-adic gamma formulation and a modular-form formulation (Guo et al., 2019).

The $\Gamma_p$3 case is central because it sits exactly at the meeting point of Ramanujan-type formulas for $\Gamma_p$4, truncated $\Gamma_p$5-type series, and $\Gamma_p$6-adic gamma phenomena. For $\Gamma_p$7,

$\Gamma_p$8

and Swisher strengthened this to modulus $\Gamma_p$9 (Liu et al., 2022).

Residue classes mod $1/\pi$0 or mod $1/\pi$1 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 $1/\pi$2 and $1/\pi$3 for $1/\pi$4 from lower-order vanishing congruences to explicit modulo-$1/\pi$5 formulas, revealing previously hidden $1/\pi$6-adic structure (Liu, 2018). 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 $1/\pi$7-quotients and expands $1/\pi$8-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 $1/\pi$9-values; a bridge lemma replaces them by $p$0-quotients; and Taylor expansions of $p$1 expose cancellations of low-order terms, producing the “super” modulus (Long et al., 2014). Their expansion

$p$2

is the local analytic engine behind many refinements (Long et al., 2014).

A second line uses the Wilf–Zeilberger method. In the proof of $p$3, 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 (Osburn et al., 2015). More recently, a systematic “streamlined WZ” framework has been developed: suitably chosen WZ devices, followed by Long–Ramakrishna $p$4-adic gamma approximations, furnish uniform proofs of $p$5, $p$6, $p$7, $p$8, $p$9, $1/\pi$0, and $1/\pi$1, while $1/\pi$2 becomes a special case where Gosper’s algorithm already succeeds (Valloud, 1 Aug 2025).

A third line is explicitly combinatorial. Liu’s refinements of $1/\pi$3 and $1/\pi$4 for $1/\pi$5 were derived from hypergeometric identities combined with Sigma-discovered binomial-harmonic identities and $1/\pi$6-adic gamma manipulations (Liu, 2018). Guo and Wang’s refinements of $1/\pi$7, $1/\pi$8, and two Swisher congruences use a more general WZ pair and identify Euler polynomials $1/\pi$9 as the universal correction term at the $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$0-level (Guo et al., 16 Jan 2025).

Despite the success of these methods, the literature repeatedly emphasizes a conceptual gap: even after the last original case $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$1 was proved, there was still no known general framework explaining why Ramanujan-type series should systematically produce supercongruences of the observed strength (Osburn et al., 2015).

4. $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$2-analogues and cyclotomic deformation

A large modern branch of the subject studies $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$3-supercongruences, where ordinary factorials are replaced by $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$4-shifted factorials,

$$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$5

ordinary integers by $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$6-integers

$$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$7

and powers of $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$8 by powers of cyclotomic polynomials $$\sum_{n=0}^{\infty} \frac{(1/2)_n^3}{n!^3}(42n+5)\frac{1}{64^n} = \frac{16}{\pi},$$9 or by mixed moduli such as $(\mathrm{A}.2)$00 (Wei, 2020). The heuristic is that, when $(\mathrm{A}.2)$01 is prime and $(\mathrm{A}.2)$02, congruences modulo $(\mathrm{A}.2)$03 recover congruences modulo $(\mathrm{A}.2)$04.

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 $(\mathrm{A}.2)$05-analogue whose specializations $(\mathrm{A}.2)$06 and $(\mathrm{A}.2)$07 recover $(\mathrm{A}.2)$08 and $(\mathrm{A}.2)$09, showing that a Ramanujan $(\mathrm{A}.2)$10-type congruence and a modular-form-valued congruence can be encoded by one basic-hypergeometric identity (Guo et al., 2019).

For $(\mathrm{A}.2)$11, Wei derived the missing modulus-$(\mathrm{A}.2)$12 $(\mathrm{A}.2)$13-analogue in the class $(\mathrm{A}.2)$14, complementing Guo’s earlier $(\mathrm{A}.2)$15 result and thereby covering all odd $(\mathrm{A}.2)$16 at the $(\mathrm{A}.2)$17-level (Wei, 2020). For $(\mathrm{A}.2)$18, creative microscoping produced a complete $(\mathrm{A}.2)$19-analogue modulo the fourth power of a cyclotomic polynomial, confirming a conjecture of Guo (Guo, 2019).

The $(\mathrm{A}.2)$20 line is especially developed on the $(\mathrm{A}.2)$21-side. Liu and Wang constructed two $(\mathrm{A}.2)$22-analogues of Swisher’s $(\mathrm{A}.2)$23 strengthening and a master parametric $(\mathrm{A}.2)$24-congruence modulo $(\mathrm{A}.2)$25; their formulas recover Swisher’s congruence under $(\mathrm{A}.2)$26 and $(\mathrm{A}.2)$27 (Liu et al., 2022). A further 2026 synthesis via the $(\mathrm{A}.2)$28-Zeilberger algorithm unified the $(\mathrm{A}.2)$29-analogues of $(\mathrm{A}.2)$30 and $(\mathrm{A}.2)$31, lifted the modulus from $(\mathrm{A}.2)$32 to $(\mathrm{A}.2)$33, and extracted $(\mathrm{A}.2)$34-level refinements involving Bernoulli numbers in the classical limit (Li et al., 27 Mar 2026).

The main technical motifs on the $(\mathrm{A}.2)$35-side are Watson’s $(\mathrm{A}.2)$36 transformation, Andrews’s and Jain’s $(\mathrm{A}.2)$37-Whipple formulas, the $(\mathrm{A}.2)$38-Dixon sum, Jackson’s $(\mathrm{A}.2)$39, polynomial Chinese remainder theorems for coprime factors such as $(\mathrm{A}.2)$40, and the creative microscoping paradigm introduced by Guo and Zudilin (Liu et al., 2022, Wei, 2020, Guo, 2019).

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{A}.2)$41, $(\mathrm{A}.2)$42, and two Swisher supercongruences from modulus $(\mathrm{A}.2)$43 to modulus $(\mathrm{A}.2)$44, with correction terms expressed through Euler polynomials $(\mathrm{A}.2)$45 (Guo et al., 16 Jan 2025). On the $(\mathrm{A}.2)$46-side, a refined unified $(\mathrm{A}.2)$47-analogue of $(\mathrm{A}.2)$48, $(\mathrm{A}.2)$49, and $(\mathrm{A}.2)$50 was then obtained modulo $(\mathrm{A}.2)$51, and its $(\mathrm{A}.2)$52 specialization recovers the corresponding parametric Euler-polynomial supercongruence (Wang et al., 7 May 2025).

Another direction replaces single sums by double or triple basic hypergeometric series. For kernels associated with $(\mathrm{A}.2)$53, Wei proved double- and triple-series $(\mathrm{A}.2)$54-supercongruences modulo the sixth power of a cyclotomic polynomial, and the specialization $(\mathrm{A}.2)$55, $(\mathrm{A}.2)$56 yielded classical double and triple supercongruences modulo $(\mathrm{A}.2)$57 (Wei, 2024). A related 2024 paper established further $(\mathrm{A}.2)$58-supercongruences for multiple basic hypergeometric series modulo the fifth and sixth powers of cyclotomic polynomials, including double-sum generalizations attached to $(\mathrm{A}.2)$59, Long’s supercongruence, and double/triple conclusions associated with $(\mathrm{A}.2)$60 (Wei, 2024).

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{A}.2)$61 the final case (Osburn et al., 2015). 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 (Valloud, 1 Aug 2025). Another concerns arithmetic refinements: $(\mathrm{A}.2)$62 has been pushed from modulus $(\mathrm{A}.2)$63 to $(\mathrm{A}.2)$64, and then to $(\mathrm{A}.2)$65-level expansions in the $(\mathrm{A}.2)$66-to-$(\mathrm{A}.2)$67 limit; $(\mathrm{A}.2)$68 has modulus-$(\mathrm{A}.2)$69 refinements; $(\mathrm{A}.2)$70 and $(\mathrm{A}.2)$71 now admit $(\mathrm{A}.2)$72-level Euler-polynomial corrections (Liu et al., 2022, Liu, 2018, Guo et al., 16 Jan 2025).

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 (Guillera, 2010).

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, $(\mathrm{A}.2)$73-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, $(\mathrm{A}.2)$74-deformations, parametric families, and higher-dimensional extensions (Osburn et al., 2015, Long et al., 2014).