An Explicit Eventual p⁴-Divisibility Theorem for an Apery-Type Numerator Family

1. Introduction

The Apery numbers and their relatives are a central source of striking congruences and supercongruences in number theory. OEIS A357513 is attached to the numerators of a rational Apery-type binomial sum,

$$a\left(n\right)=num\left({\displaystyle \sum _{k=1}^n}{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0pt}{}{n+k}{k}\right)}^2}{k^3}}\right),$$

with $a\left(0\right)=0$. The same source records a broader m-parameter finite-exception conjecture: for a nonnegative integer m, if

$$u_m\left(n\right)=num\left({\displaystyle \sum _{k=1}^n}{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0pt}{}{n+k}{k}\right)}^2}{k^{2m+1}}}\right),$$

then $u_m(p-1)\equiv 0(mod{p}^4)$ for all primes p, apart from finitely many exceptions depending on m. The source formulation is qualitative: it predicts finite exception sets but does not give a uniform construction of such a set.

The main theorem establishes this statement in an explicit eventual form. For every nonnegative integer m, every prime $p>2m+6$ satisfies

$$u_m(p-1)\equiv 0(mod{p}^4).$$

Equivalently, one may take the exceptional set to be $\{0,1,\dots ,2m+6\}$. This is a sufficient bound produced by the argument, not a claim that these are the sharp exceptional primes for a given m.

The mathematical contribution proved here is therefore an explicit uniform eventual bound for this numerator family. The result upgrades the qualitative finite-exception assertion to a theorem with a concrete witness. Since the conjectural formulation is recorded on OEIS, it is natural to state the result relative to that source; however, the theorem itself should be read as an effective supercongruence statement, while any absolute claim about first appearance in the literature depends on the broader related-work discussion rather than on the OEIS record alone.

The proof is short conceptually but sensitive to denominator bookkeeping. Working in $\mathbb{Z}/p^4\mathbb{Z}$, where $1,\dots ,p-1$ are units, we expand the squared binomial product in the summand for $n=p-1$ and obtain the congruence

$${\left({\displaystyle \genfrac{}{}{0pt}{}{p-1}{k}}\right)}^2{\left({\displaystyle \genfrac{}{}{0pt}{}{p-1+k}{k}}\right)}^2{k}^{-(2m+1)}\equiv p^2{k}^{-(2m+3)}-2p^3{k}^{-(2m+4)}(mod{p}^4).$$

Thus the full sum reduces to two inverse-power sums. The even-exponent term vanishes by the standard finite-field power-sum identity, and the odd-exponent term vanishes by a mod-$p^2$ pairing of k with $p-k$. A final common-denominator lemma then transfers the resulting residue computation for the rational sum to the numerator in lowest terms.

2. Source and Related Work

The primary source is OEIS A357513 [1]. That entry records the base sequence, the special $m=1$ supercongruence, and the general finite-exception conjecture. The page currently marks the $m=1$ case as true via a linked AlphaProof/FormalConjectures proof, while the general m-parameter statement is still written there as a conjecture. The theorem established here supplies an explicit eventual bound for that general OEIS statement, with $E_m=\{0,1,\dots ,2m+6\}$.

There is one source convention to fix. The OEIS general comment writes the generalized sum as starting with $k=0$, but the factor $1/k^{2m+1}$ is undefined at $k=0$. The FormalConjectures source file records this as an apparent typo and uses the mathematically meaningful range $k=1,\dots ,n$ [2]. The theorem in this paper uses that same convention.

The surrounding mathematical context is the theory of Apery-type sequences and their supercongruences. The kernel ${\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}^2{\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)}^2$ is the classical Apery kernel, and congruence phenomena for Apery-like sequences have been studied in several settings; see, for instance, the work of Osburn and Sahu on Apery-like numbers and Straub’s multivariate perspective on supercongruences of rational functions [3,4]. More recent and adjacent frameworks include binomial-coefficient harmonic-sum identities used in supercongruence proofs [5] and Beukers-type methods for families of truncated hypergeometric and Apery-like congruences [6]. These works show that many congruences near the Apery kernel can be treated by broad methods.

The present theorem is deliberately narrower. It is not a new general framework for Apery-type supercongruences. Rather, it computes the specific numerator sum above at $n=p-1$ directly modulo $p^4$, including the passage from a rational congruence to the numerator in lowest terms. The cited general frameworks do not, in the forms reviewed here, state the exact m-parameter numerator family or the explicit eventual exception set $E_m=\{0,1,\dots ,2m+6\}$.

At the p-adic level, the proof also belongs to the circle of Wolstenholme-type congruences for harmonic and inverse power sums. The reduction of the binomial factor to inverse-power sums, and the subsequent cancellation $mod{p}^2$ and $modp$, are elementary instances of techniques that are standard in the modern literature on harmonic-sum congruences and multiple harmonic sums [7,8]. In particular, the odd-pairing step used below is of the same flavor as the classical congruence $H_{p-1}\equiv 0(mod{p}^2)$ and its higher-power analogues.

As for the exact numerator family

$$num\left({\displaystyle \sum _{k=1}^n}{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0pt}{}{n+k}{k}\right)}^2}{k^{2m+1}}}\right),$$

the explicit formulation used here appears in OEIS A357513 and in the linked FormalConjectures source [1,2]. We are not aware of an earlier publication devoted specifically to this family or to the eventual bound $p>2m+6$. Accordingly, the claim of the paper is the proof of that explicit eventual bound, not a broader priority claim about Apery-type supercongruences in general and not a claim that other methods could not also prove the same statement.

3. Statement

For a rational number q, write $num\left(q\right)$ for the absolute value of the numerator of q in lowest terms. This convention is sign-insensitive for the congruence $num\left(q\right)\equiv 0(mod{p}^4)$. For $m,n\ge 0$, define

$$u_m\left(n\right)=num\left({\displaystyle \sum _{k=1}^n}{\displaystyle \frac{{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0pt}{}{n+k}{k}\right)}^2}{k^{2m+1}}}\right).$$

Theorem 1.

Let $m\ge 0$. If p is a prime and $p>2m+6$, then

$$u_m(p-1)\equiv 0(mod{p}^4).$$

Consequently, the OEIS A357513 finite-exception conjecture holds with the explicit exception set

$$E_m=\{0,1,\dots ,2m+6\}.$$

The final passage from the rational Apery-type sum to the reduced numerator is isolated in the following elementary lemma, which will be applied after the residue computation modulo $p^4$.

Lemma 1.

Let $q=N/D\in \mathbb{Q}$ with $N,D\in \mathbb{Z}$, $D\ne 0$, and $gcd(D,p)=1$. If $N\equiv 0(mod{p}^4)$, then

$$num\left(q\right)\equiv 0(mod{p}^4).$$

Indeed, if $g=gcd(N,D)$, then $q=(N/g)/(D/g)$ in lowest terms up to the harmless sign convention for the denominator. Since $g\mid D$ and $gcd(D,p)=1$, the factor g is a unit modulo $p^4$. Thus divisibility of N by $p^4$ is unchanged when one passes from the unreduced numerator N to the reduced numerator $N/g$, and taking absolute value does not affect congruence to 0.

The archived formal artifact checked for this result is the result-bundle release dated 2026-06-10, whose verified Lean module is

AmraLibrary/OpenProblemBatches/TrueOpenNextRound20260606/04generalsupercongruencezmodcast.lean.

The build report records a successful lake env lean verification of the declarations

theorem general_supercongruence_source_statement

    (m : Nat) : exists exceptions : Finset Nat, forall p, p.Prime ->

      p notin exceptions -> OeisA357513.u m (p - 1) = (0 : ZMod (p ^ 4))

theorem general_supercongruence_eventual

    (m : Nat) :

    exists exceptions : Finset Nat, forall p, p.Prime ->

      p notin exceptions ->

      ((OeisA357513.u m (p - 1) : Nat) : ZMod (p ^ 4)) = 0

Both displayed Lean theorem types are existential in the exception set, because the public source-facing interface follows the FormalConjectures shape. The effective bound in thm:a357513 is formalized inside the proof term of general_supercongruence_eventual: its witness for the existential exception set is $\mathtt{Finset}.\mathtt{Icc}\mathtt{0}(\mathtt{2}*\mathtt{m}+\mathtt{6})$. The source-facing wrapper then obtains the source statement from that effective theorem. Thus the printed Lean type is existential, while the Lean proof object supplies the explicit finite set used in the manuscript.

4. Proof

The proof uses two standard congruence inputs: the nontrivial power-sum vanishing over ${\mathbb{F}}_p^{\times }$, and the first-order binomial expansion ${(1-x)}^{-e}\equiv 1+ex(mod{x}^2)$ applied with $x=p/k$. The remaining work is the bookkeeping needed to reduce the Apery-type numerator sum to these two elementary facts.

Lemma 2

(Summand expansion). Let $m\ge 0$, let p be prime, and let $1\le k\le p-1$. In $\mathbb{Z}/p^4\mathbb{Z}$,

$${\left({\displaystyle \genfrac{}{}{0pt}{}{p-1}{k}}\right)}^2{\left({\displaystyle \genfrac{}{}{0pt}{}{p-1+k}{k}}\right)}^2{k}^{-(2m+1)}\equiv p^2{k}^{-(2m+3)}-2p^3{k}^{-(2m+4)}(mod{p}^4).$$

Proof. 

Put $R=\mathbb{Z}/p^4\mathbb{Z}$. Since $1\le k\le p-1$, every $1,\dots ,k$ is a unit in R. In R, the binomial coefficients admit the product expansions

$$\left({\displaystyle \genfrac{}{}{0pt}{}{p-1}{k}}\right)={(-1)}^k{\displaystyle \prod _{j=1}^k}\left(1-{\displaystyle \frac{p}{j}}\right),\left({\displaystyle \genfrac{}{}{0pt}{}{p-1+k}{k}}\right)={\displaystyle \frac{p}{k}}{\displaystyle \prod _{j=1}^{k-1}}\left(1+{\displaystyle \frac{p}{j}}\right).$$

After squaring and separating the factor with $j=k$, we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \genfrac{}{}{0pt}{}{p-1}{k}}\right)}^2{\left({\displaystyle \genfrac{}{}{0pt}{}{p-1+k}{k}}\right)}^2& =p^2{k}^{-2}{\left(1-{\displaystyle \frac{p}{k}}\right)}^2{\displaystyle \prod _{j=1}^{k-1}}{\left(1-{\displaystyle \frac{p}{j}}\right)}^2{\left(1+{\displaystyle \frac{p}{j}}\right)}^2\hfill \\ \hfill & =p^2{k}^{-2}{\left(1-{\displaystyle \frac{p}{k}}\right)}^2{\displaystyle \prod _{j=1}^{k-1}}{\left(1-{\displaystyle \frac{p^2}{j^2}}\right)}^2.\hfill \end{array}$$

Set

$$P_k={\displaystyle \prod _{j=1}^{k-1}}{\left(1-{\displaystyle \frac{p^2}{j^2}}\right)}^2.$$

Each factor satisfies ${\left(1-{\displaystyle \frac{p^2}{j^2}}\right)}^2\equiv 1(mod{p}^2)$. A finite product of elements congruent to 1 modulo $p^2$ is again congruent to 1 modulo $p^2$, so

$$P_k\equiv 1(mod{p}^2).$$

Hence $P_k=1+p^2{C}_k$ for some $C_k\in R$. Multiplying by the leading factor $p^2{k}^{-2}$, the $p^2{C}_k$ error contributes a multiple of $p^4$, and we get

$$p^2{k}^{-2}{\left(1-{\displaystyle \frac{p}{k}}\right)}^2{P}_k\equiv p^2{k}^{-2}{\left(1-{\displaystyle \frac{p}{k}}\right)}^2(mod{p}^4).$$

Now

$${\left(1-{\displaystyle \frac{p}{k}}\right)}^2=1-{\displaystyle \frac{2p}{k}}+{\displaystyle \frac{p^2}{k^2}},$$

so

$$p^2{k}^{-2}{\left(1-{\displaystyle \frac{p}{k}}\right)}^2=p^2{k}^{-2}-2p^3{k}^{-3}+p^4{k}^{-4}\equiv p^2{k}^{-2}-2p^3{k}^{-3}(mod{p}^4).$$

Multiplying by $k^{-(2m+1)}$ yields the stated congruence. □

Lemma 3

(Inverse-power cancellation). Let $m\ge 0$, let p be prime, and suppose $p>2m+6$. In $\mathbb{Z}/p^4\mathbb{Z}$, set

$$H_a={\displaystyle \sum _{k=1}^{p-1}}{k}^{-a}.$$

Then

$$H_{2m+4}\equiv 0(modp)\mathrm{and}{H}_{2m+3}\equiv 0(mod{p}^2).$$

Proof. 

Let $e=2m+3$. The hypothesis $p>2m+6$ is used here to obtain

$$0<e+1=2m+4<p-1.$$

Reduction modulo p identifies $H_{e+1}$ with

$$\sum _{x\in {\mathbb{F}}_p^{\times }}{x}^{-(e+1)}.$$

Because inversion permutes ${\mathbb{F}}_p^{\times }$, this equals ${\sum }_{x\in {\mathbb{F}}_p^{\times }}{x}^{e+1}$. The multiplicative group ${\mathbb{F}}_p^{\times }$ is cyclic of order $p-1$, so this power sum is 0 whenever $0<e+1<p-1$. Hence

$$H_{e+1}\equiv 0(modp).$$

For the odd-exponent sum, the map $k\mapsto p-k$ permutes $\{1,\dots ,p-1\}$. Therefore

$$2H_e={\displaystyle \sum _{k=1}^{p-1}}\left(k^{-e}+{(p-k)}^{-e}\right).$$

Now e is odd, so in $\mathbb{Z}/p^2\mathbb{Z}$ we have

$$\begin{array}{cc}\hfill {(p-k)}^{-e}& ={(-k)}^{-e}{\left(1-{\displaystyle \frac{p}{k}}\right)}^{-e}\hfill \\ \hfill & \equiv -k^{-e}\left(1+{\displaystyle \frac{ep}{k}}\right)(mod{p}^2),\hfill \end{array}$$

because terms of order ${(p/k)}^2$ vanish modulo $p^2$. Therefore

$$k^{-e}+{(p-k)}^{-e}\equiv -ep{k}^{-(e+1)}(mod{p}^2),$$

and summing gives

$$2H_e\equiv -ep{H}_{e+1}(mod{p}^2).$$

Since $H_{e+1}\equiv 0(modp)$, the right-hand side is 0 modulo $p^2$. Because $p>2m+6\ge 6$, the prime p is odd and 2 is a unit modulo $p^2$. Thus $H_e\equiv 0(mod{p}^2)$. Substituting $e=2m+3$ proves both claims. □

Proof of Theorem 1.

Fix $m\ge 0$ and a prime $p>2m+6$. Put

$$R=\mathbb{Z}/p^4\mathbb{Z}.$$

For $1\le k\le p-1$, the residue class of k is a unit in R. Thus every occurrence of $k^{-a}$ below is well-defined in R.

Define the hypergeometric residue sum

$$S_m\left(p\right)={\displaystyle \sum _{k=1}^{p-1}}{\left({\displaystyle \genfrac{}{}{0pt}{}{p-1}{k}}\right)}^2{\left({\displaystyle \genfrac{}{}{0pt}{}{p-1+k}{k}}\right)}^2{k}^{-(2m+1)}.$$

By Lemma 2,

$$S_m\left(p\right)=p^2{H}_{2m+3}-2p^3{H}_{2m+4}\mathrm{in}R,$$

where $H_a={\sum }_{k=1}^{p-1}{k}^{-a}$. By lem:inversepowercancellation, $H_{2m+3}\equiv 0(mod{p}^2)$ and $H_{2m+4}\equiv 0(modp)$. Hence both terms on the right vanish in R, and therefore

$$S_m\left(p\right)=0\mathrm{in}R.$$

It remains to pass from the residue sum to the reduced numerator. Apply lem:numeratortransferstatement to

$$r=2m+1,A_k={\left({\displaystyle \genfrac{}{}{0pt}{}{p-1}{k}}\right)}^2{\left({\displaystyle \genfrac{}{}{0pt}{}{p-1+k}{k}}\right)}^2,D={\displaystyle \prod _{k=1}^{p-1}}{k}^r.$$

Then D is prime to p, hence a unit in R, and

$${\displaystyle \sum _{k=1}^{p-1}}{\displaystyle \frac{A_k}{k^r}}={\displaystyle \frac{N}{D}},N={\displaystyle \sum _{k=1}^{p-1}}{A}_k{\displaystyle \frac{D}{k^r}}.$$

Here each quotient $D/k^r$ is an integer because $k^r$ is one factor of the product defining D. Let a bar denote reduction modulo $p^4$. Since $D=k^r(D/k^r)$ and ${\overline{k}}^r$ is a unit in R, we have

$$\overline{D/k^r}=\overline{D}{\left({\overline{k}}^r\right)}^{-1}.$$

Therefore

$$\begin{array}{cc}\hfill \overline{N}& ={\displaystyle \sum _{k=1}^{p-1}}{\overline{A}}_k\overline{D/k^r}\hfill \\ & =\overline{D}{\displaystyle \sum _{k=1}^{p-1}}{\overline{A}}_k{\left({\overline{k}}^r\right)}^{-1}\hfill \\ & =\overline{D}{S}_m\left(p\right).\hfill \end{array}$$

The residue sum was already shown to be $S_m\left(p\right)=0$ in R, so $N\equiv 0(mod{p}^4)$. Since D is prime to p, the numerator-transfer lemma applies to the rational sum $N/D$. Its reduced numerator, with the absolute-value convention of num, is therefore also 0 modulo $p^4$. This reduced numerator is exactly $u_m(p-1)$, proving the congruence.

Finally, take $E_m=\{0,1,\dots ,2m+6\}$. If p is prime and $p\notin E_m$, then $p>2m+6$, so the congruence just proved applies. This proves the finite-exception statement. □

5. Formal Verification

The argument is backed by a Lean 4 formalization in the AMRA library, written in the Lean/mathlib environment of [9,10]. The formal proof support package is

https://github.com/chainstart/formal_proof/tree/main/a357513_general_supercongruence

at formal-proof repository commit

b7351db516d97340a8b967a5330eef9e80d5fbf5.

The support package contains the Lean source tree, reproduction instructions, verified-declaration manifest, build report, original FormalConjectures source file, source-status audit, and source archive. The source archive SHA256 checksum is

0a4b5f568528eaf2fcdc781788f16f73aaf16ff6aa649d9537ab9c9a75fd497d.

The target module is

AmraLibrary/OpenProblemBatches/TrueOpenNextRound20260606/04generalsupercongruencezmodcast.lean.

The archived target-file SHA256 checksum is

eec4e116aa4e1c8083868b6f3ccb25100be59cffffe1b328ca514cfb1064e654.

For artifact review, the support package above is the intended supplementary object. A reader can verify the archive by running sha256sum on the zip file and can check the Lean source directly from the package root with the command below. The theorem names quoted in thm:a357513 are the declarations checked by that command. The recorded verification command is

timeout 1200s prlimit--as=22000000000--env LEANNUMTHREADS=1 OMPNUMTHREADS=1 lake env lean AmraLibrary/OpenProblemBatches/TrueOpenNextRound20260606/04generalsupercongruencezmodcast.lean.

The corresponding build report records status $\mathtt{passed}$, and the ARA support manifest classifies the target declarations as $\mathtt{verified}$. The placeholder scan reports zero occurrences of sorry, admit, axiom, constant, and other placeholder markers in the target file. The checked toolchain is:

Lean 4.26.0, Lean commit d8204c9fd894f91bbb2cdfec5912ec8196fd8562;

Lake 5.0.0-src+d8204c9;

mathlib v4.26.0, revision 2df2f0150c275ad53cb3c90f7c98ec15a56a1a67;

AMRA git head e4e339e5b380375cf1c7838251966d0fc3c06929.

The AMRA checkout observed during extraction was not clean, so the source archive checksum above is the immutable reproduction identifier.

The formal development separates the effective theorem from the source-facing wrapper. The declaration

OeisA357513NextRound20260606.general_supercongruence_eventual

has an existential theorem type, but its proof body supplies the explicit witness

Finset.Icc 0 (2 * m + 6).

Thus the bound $E_m=\{0,1,\dots ,2m+6\}$ is formalized as the witness used to prove the effective Lean theorem, even though the printed theorem type follows the existential finite-exception shape. By contrast,

OeisA357513NextRound20260606.general_supercongruence_source_statement

is the wrapper matching the FormalConjectures source format: it asserts only the existence of a finite exception set for each m, and obtains that statement by exact specialization of the effective theorem. This is the same convention used in the statement section: the public theorem type is existential, while the explicit set in thm:a357513 is the witness constructed in the Lean proof.

The main formal lemmas mirror the handwritten proof. They carry out, respectively, the reduction of the Apery-type summand modulo $p^4$, the cancellation of the inverse-power sums, and the transfer from the rational hypergeometric sum to the numerator $u_m(p-1)$. The final theorem in the manuscript is therefore aligned with a concrete Lean proof object rather than only with an informal source annotation.

The formal theorem matches the manuscript at the level relevant to the source problem: it proves the finite-exception congruence for the same $u_m(p-1)$ family and supplies the explicit witness Finset.Icc 0 (2 * m + 6). The theorem is not a sharp-exception classifier, and the Lean statement deliberately retains the source-facing existential wrapper so that it remains aligned with the FormalConjectures interface.

6. Scope and Limitations

thm:a357513 is an eventual theorem. It proves the finite-exception statement for the generalized numerator congruence and supplies the explicit bound $p>2m+6$, equivalently the exception set $\{0,1,\dots ,2m+6\}$, but it does not determine the minimal exceptional set for each fixed m. That distinction is genuine: the OEIS entry records sharper behavior in special cases, so the theorem should be read as an effective uniform bound rather than as a sharp classification of all exceptions.

The bound is also not expected to be sharp in small m. For example, the FormalConjectures source records the $m=1$ specialization as holding for all primes $p\ge 3$ except $p=7$, whereas the uniform theorem here only gives the automatic threshold $p>8$. The proof deliberately favors a uniform inequality, $0<2m+4<p-1$, that makes the finite-field power-sum cancellation immediate for every m. Refining the small exceptional primes would require additional case analysis and is outside the purpose of this note.

The result also sits in a broader literature on Apery-type supercongruences and Wolstenholme-type harmonic-sum congruences [3,4,7,8]. Its specific contribution is narrower than those general theories. The argument treats the concrete numerator family attached to OEIS A357513 by an explicit modulo-$p^4$ computation at $n=p-1$; it does not attempt a wider classification of related Apery kernels or a multivariate framework.

The source page separately marks the $m=1$ case as proved by the linked AlphaProof and FormalConjectures artifact. The present theorem is therefore not merely a reformulation of that isolated case. Rather, it establishes the full m-parameter eventual statement for every nonnegative m, for all primes sufficiently large relative to m.

The provenance claim should remain correspondingly cautious. OEIS A357513 records the general statement as a conjecture, and the source audit bundled with this manuscript did not locate a public proof of the full general theorem among the indexed sources checked. This supports the claim that the paper proves an explicit eventual bound for the OEIS conjectural family. It does not by itself justify an absolute first-priority claim, which should be reserved until a broader independent literature check has been completed.

7. Conclusions

We established an explicit eventual supercongruence for the numerator sequence associated with OEIS A357513: for each nonnegative integer m, the stated congruence holds for every prime $p\notin E_m=\{0,1,\dots ,2m+6\}$, equivalently every prime $p>2m+6$. The argument proceeds by a mod-$p^4$ expansion of the Apery-type binomial summand, the resulting inverse-power-sum cancellations, and a final transfer from the rational congruence to the numerator in lowest terms.

Accordingly, the main contribution of the paper is the proof of this explicit finite exceptional bound together with a formal verification of the source-facing theorem in the AMRA Lean development. The proof closes the general eventual statement, while leaving the sharper problem of determining minimal exceptional sets for fixed m. Questions of priority relative to the broader literature on Apery-type supercongruences and related numerator families should be read in that narrower scope.