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Claim. For all sufficiently large $n$ there are residue classes $a_p\pmod p$, one for each prime $p\le n$, such that every integer in $[1,n]$ lies in at least two of them: a yes to Problem 689. The claimant is Malek Zribi, posting under the account MalekZ. The strategy, as the first posting describes it: take the odd class modulo $2$ and the zero class modulo $3$, move a fixed finite set $S$ of small primes to nonzero classes, so that the integers still short of two hits are, in the main, even numbers $2^kuq$ with $u$ odd and $S$-smooth and $q$ a prime outside $S$, about $(1+o(1))n/\log n$ of them by the prime number theorem in progressions; call a prime $P>n/5$ robust when the switched classes already give its multiples $P$, $2P$ and $4P$ enough hits, so that moving $P$ to a new class creates no new shortfall; build a $3$-partite hypergraph on two cores of shortfall integers and the robust primes $P$ in $(n/5,\beta n]$, with an edge $(x,y,P)$ when $|y-x|=2P$, so that one class modulo $P$ covers both $x$ and $y$; obtain a fractional matching from first and second moments supplied by the linear-equations-in-primes theorem of Green, Tao and Ziegler; round it to a genuine matching by Kahn's theorem on fractional matchings; and cover the leftover singletons with the unused robust primes, whose surplus a numerical margin guarantees. The notes make precise the route sketched earlier in the thread by Sawhney and Tao.

Submission note. Posted to the site's forum by Malek Zribi on 25 April 2026:

I have a proposed proof of Problem 689 for all sufficiently large $n$. A short PDF note is here:

PDF note

Disclosure: this write-up was prepared with AI assistance in finding literature and reviewing claims. I am posting it as a proposed proof and request for verification, not as a refereed result.

This is meant as an attempt to make precise the route suggested in the earlier comments: use linear equations in primes to control averaged prime-pattern counts, and use hypergraph matching/nibble technology to turn the resulting fractional cover into a genuine disjoint cover. The new ingredient in the note is the robust $P>n/5$ cleanup setup and the finite-core fractional matching formulation.

The strategy is as follows.

Start with $a_2\equiv 1\pmod 2$, leave $3$ in the zero class, and switch a fixed finite set

$$S\subset\{7,11,13,\ldots\}$$

to nonzero

residues. After these fixed switches, the main residual demand consists of even numbers

$$x=2^k u q,\qquad k\ge1,$$

where $u$ is odd $S$-smooth

and $q\notin S$ is prime, subject to fixed congruence exclusions. Fixed-modulus PNT gives

$$|A_S(n)|=(1+o(1))\frac n{\log n},$$

with half of

this mass at $v_2(x)=1$ and half at $v_2(x)\ge2$.

The cleanup primes are primes $P>n/5$. Call $P$ robust if

$$> H_S(P)\ge1,\qquad H_S(2P)\ge2,\qquad H_S(4P)\ge2,$$

where $H_S(x)$ counts

the switched $S$-classes hitting $x$. Switching a robust $P$ creates no new unresolved debt: the only multiples $P,2P,3P,4P\le n$ are covered by robustness plus parity and the unchanged $0\pmod3$ class. By choosing $S$ large enough, the robust residue density $\delta_S$ can be made $>0.94388\ldots$.

Choose

$$\delta_S^{-1}-3/5<\beta<\tfrac12(1-\tfrac35 e^{-2}),$$

and set

$\Delta=(\beta+3/5)\delta_S-1>0$ for the surplus margin used below. Build a 3-partite hypergraph with vertex classes

$$X=X_n\subset A_1(n),\qquad > Y=Y_n\subset A_2(n),\qquad Z=\{P\in(n/5,\beta n]:P\text{ robust}\},$$

where

$X_n,Y_n$ are finite coefficient cores capturing fractions $\alpha_X,\alpha_Y>G(\beta)+\eta$ of the $A_1,A_2$ coefficient mass, and edges

$$(x,y,P)\quad\text{when}\quad |y-x|=2P.$$

A matching covering

$(1-o(1))|Z|$ labels gives enough paired covers; the remaining residual targets, including the coefficient tails outside the finite cores and the $o(n/\log n)$ exceptional residual tokens, are covered singly by unused robust primes. The strict surplus $\Delta>0$ is exactly what leaves enough unused robust primes for the singleton cleanup once the cores are chosen so that the discarded coefficient-tail mass plus exceptional tokens is at most $\Delta N/10$.

The matching is obtained by a finite-core fractional construction. In half-residue coordinates $A\equiv aq\pmod W$, $B\equiv bq'\pmod W$, the residual classes are a product set $\mathcal C$, and for every unit label residue $\pi$,

$$\#\{A\in\mathcal C:A\pm\pi\in\mathcal C\}=\prod_{s\in > S}(s-2).$$

This gives an explicit continuum transport kernel with exact label

load $1$ and side load bounded by

$$> G(\beta)=\int_{1/5}^{\beta}\frac{dt}{1-2t}<1.$$

Finite coefficient cores scale

this by the captured core mass, giving side bounds $G(\beta)/\alpha_X$ and $G(\beta)/\alpha_Y$, both strictly less than $1$ by the choice of cores. The aggregate transport then lifts to bounded typed kernels on the finitely many admissible typed polygons, with limiting label load $1$ and bounded side loads.

The analytic input is the finite-complexity Green-Tao-Ziegler theorem for affine-linear forms in primes. In a detailed proof this can be formulated using an auxiliary growing $W_{\mathrm{GTZ}}$-trick while keeping the fixed modulus $W=\prod_{s\in S}s$ as residue data; equivalently, one can work with fixed-modulus singular series and verify the corresponding local-factor disintegrations in the first and second moment systems. It supplies the first and second weighted moment estimates for the edge loads. Importantly, no pointwise Hardy-Littlewood / Bateman-Horn estimate for fixed $P=bq'-aq$ is used; all estimates are averaged finite-complexity linear-form counts.

The deletion step uses the standard $L^2$-to-mass-loss argument: vertices with normalized side load above $1$ have (|L_X(x)-L_X^{\mathrm{lim}}(x)|\ge 2\gamma), so the side $L^2$ estimate forces $|B_X|=o(|X_n|)$, and a Cauchy-Schwarz finish bounds the deleted mass by $o(|Z_n|)$ on each side.

The final rounding input is Kahn's fractional Frankl-Rödl-Pippenger theorem (Random Structures and Algorithms 8 (1996), 149-157), applied with the single statistic $C(e)=1$. The hypergraph has codegree at most $2$: a pair $(x,y)$ determines $P=|y-x|/2$, and a pair $(x,P)$ or $(y,P)$ has at most two extensions. Hence

$$a(t)=\max_{u\ne v}\sum_{e\supset\{u,v\}}t_e\le > 2\max_e t_e=o(1).$$

What I would especially appreciate help with.

The two interfaces I would most like checked are:

(1) Kahn 1996, Theorem 1.5. I have not been able to access the printed paper directly. Public metadata/abstracts for Kahn's paper confirm the pair co-load parameter

$$\alpha(t)=\max\Big\{\sum\{t(A): x,y\in A\in\mathcal > H\}:x,y\in V,\ x\ne y\Big\}$$

exactly as I use it. What I cannot verify from

public sources is the precise form of the conclusion in the non-perfect case: specifically, that for a fractional matching $t$ with total mass (\sum_e t_e=(1-o(1))|Z_n|), $\max_e t_e=o(1)$, and $\alpha(t)=o(1)$, Theorem 1.5 produces an integral matching $M$ with $|M|=(1-o(1))|Z_n|$. If anyone with access to the printed paper can confirm that this is the right shape of the conclusion, or flag a hypothesis I have missed, I would be very grateful.

(2) The GTZ moment formulation. I use one edge-total system and three second-moment systems on a fixed finite coefficient core. The note identifies the linear forms used in each system and verifies that no two are rationally affinely dependent after diagonal removal. The local-factor disintegration of the second-moment main terms, under either an auxiliary $W_{\mathrm{GTZ}}$-trick or fixed-modulus singular series, is asserted rather than written out in full. I would welcome any flag if this is not the right shape, or if the systems require additional admissibility checks I have missed.

I do not currently see a hidden pointwise prime-pair input, no Hardy-Littlewood, Bateman-Horn, Elliott-Halberstam, or Goldbach-type pointwise estimate is used in the argument as written though I would welcome being shown otherwise.

Posted to the site's forum by Malek Zribi on 26 April 2026:

Updated PDF with Kahn's Theorem 1.5 verified directly against the printed paper, the typed-kernel lift written out with explicit load identities, and the GTZ admissibility and local-factor identities checked in line: pdf (prepared with AI assistance, as before, specifically from Claude and codex). I'd very much appreciate a careful read of the second-moment systems (22)–(24) and the local-factor identities (26), as these are the steps where independent eyes would matter most.

Posted to the site's forum by Malek Zribi on 29 April 2026:

I’ve rewritten the proposed proof as a verification package, with the deterministic debt/matching steps separated from the analytic input. The only point I’m asking people to check now is whether the weighted GTZ moment proposition in Appendix A really follows from the finite-complexity linear-equations-in-primes theorem. In particular, are the four local-factor identities for the edge, Z-, X-, and Y-second-moment systems correct? AI tools helped with the exposition/checking, but I have personally read the note and verified it to my highest degree, simply asking for whats left (if anything) for a verified solution. Thanks in advance to anyone who decides to review. PDF note here

Posted to the site's forum by Malek Zribi on 2 June 2026:

Here's my attempted solution revolving on the reduction of problem to a finite-core prime-difference matching I simply need some input on the construction of the Green–Tao’s codimension-≤2 affine-linear prime theorem, plus Kahn’s fractional matching theorem. This version includes an explicit local-factor convention and moment audit. I appreciate any help on this, and whether or not this fully closes the problem after a good check. I used 5.5 to tidy up some portions of the work. This proof builds off of Premzek's previous note along side my own prior work on this problem. PDF

Postings. The notes are four PDFs shared through Google Drive, one per posting, linked above as the preprint links in the order of the thread posts that carry them; this page records the claim from the posts' own descriptions. 25 April 2026: the first note, which the author calls a proposed proof and a request for verification, not a refereed result, prepared with AI assistance in finding literature and reviewing claims; a thread reader replied the next day that the strategy was interesting but at best a sketch, and the author agreed that a proof sketch and verification request is the right description. 26 April 2026: an updated note checking Kahn's theorem against the printed paper and writing out the load identities and the local factors, with a request that the second-moment systems and the local-factor identities be read, prepared, the post says, with assistance from Claude and Codex. 29 April 2026: a rewrite as a verification package separating the deterministic debt-and-matching steps from the analytic input, asking only whether the weighted moment proposition follows from the finite-complexity linear-equations theorem. 2 June 2026: an attempted solution reducing the problem to a finite-core prime-difference matching, building on Chojecki's manuscript and the author's own earlier notes, with a model the post names only as 5.5 used to tidy parts of the text; a thread reader called it a full solution candidate on which a standard check had found no issue, a forum check and not a review, and the author accepted that description. The April and June notes were then merged with Chojecki's additions into the joint submission on the proof-claim tab, which supersedes them as the claimants' current text.

Dating. The page carries the date of the first posting, as the corpus dates claims. The author agreed on 26 April 2026 that the first note was a proof sketch and verification request, a description below a proof, but did not withdraw the argument or retitle it as a reduction: the postings of 26 and 29 April revise the same argument, the posting of 2 June 2026 presents it as an attempted solution, and the joint submission of 21 July 2026 asserts it as a full proof. The claim is recorded as the one argument developed through these postings, with the 2 June note as the first text its author presents as a solution; the standing below rests on that posting and on the joint submission, not on the April sketches.

Standing. Claimed. The site's label is OPEN (page last edited 8 April 2026; thread as of 2026-10-07). On 2 June 2026 the site's maintainer pinned a comment saying that two full-solution claims, this one and Chojecki's, had now been posted, both built on the sketch developed in the thread mainly by Sawhney and Tao, that the maintainer would wait for a refereed publication or a careful reading by an expert before changing the label, and that further AI-generated elaborations of that sketch should be discussed elsewhere. No refereed version, independent review or site acceptance was found on 2026-10-07.

Depends on. Chojecki's manuscript, on whose note the author says the text of 2 June 2026 builds; the April notes rest on no page of this wiki.