Problems 66 and 67 on sums of residue classes and primes of Andr\'{a}s S\'{a}rk\"{o}zy's collection of unsolved problems

Christian Elsholtz; Yuchen Ding; Yu-Chen Sun

REVIEW 3 major objections 5 minor 20 references

Problems 66 and 67 on sums of residue classes and primes of Andr\'{a}s S\'{a}rk\"{o}zy's collection of unsolved problems

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper settles two open problems: it gives counterexamples to both parts of the residue-class sumset problem, and it constructs a prime set of lower relative density 5/8 that leaves infinitely many odd integers unreachable as…

desk verdict A sharp 5/8 counterexample for the density three-prime problem; the proof is sound and the result is worth publishing. read the letter →

arxiv 2508.02433 v1 pith:TH4ECKWG submitted 2025-08-04 math.NT

Yuchen Ding , Christian Elsholtz , Yu-Chen Sun This is my paper · ORCID

classification math.NT MSC 11A4111A0511P32

keywords three-primesumproblemssumsetsprimesinarithmeticprogressionscongruenceobstacleresidueclassesdensityversionofthetheoremrelativeprimesparsesubsets

topics Goldbach's Conjecture

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

The paper answers two open problems from a collection of unsolved problems in number theory. For Problem 66, about sumsets of reduced residue classes modulo $m$, it shows that both parts fail as stated: there exist sets $A$ with $|A| > (\tfrac{1}{2}+\varepsilon)\varphi(m)$ whose two-fold or three-fold sumset misses a positive proportion of even or odd residue classes. It also records that if the hypothesis is raised to $|A| > \tfrac{5}{8}\varphi(m)$ and $m$ is odd and square-free, the ternary statement becomes true, so the corrected version is sharp. For Problem 67, the main theorem constructs an infinite set of primes $Q$ with upper relative density $1$ and lower relative density exactly $\tfrac{5}{8}$ such that infinitely many odd integers cannot be written as a sum of three elements of $Q$. This proves that the $\tfrac{5}{8}$ threshold in the density version of the three-prime theorem is best possible.

What carries the argument

The construction alternates two types of prime blocks: odd-indexed blocks contain all primes in intervals $(x_{2k-1},x_{2k}]$, which pushes the upper relative density to $1$, and even-indexed blocks contain only primes congruent to an element of $A_1=\{1,2,4,7,13\}$ modulo $15$, which contributes the lower density $\tfrac{5}{8}$ via the standard count of primes in arithmetic progressions. The load-bearing estimate is a sieve upper bound, equation (5), of the form $t_p \ll x_{2k+1}(\log\log x_{2k+1})/(\log x_{2k+1})^2$ for the number $t_p$ of representations of $x_{2k+1}-p$ as a sum of two primes; it ensures that only $O(x_{2k+1}(\log\log x_{2k+1})^2/(\log x_{2k+1})^2)$ primes are deleted. The 'trap' integers are the $x_{2k+1}$, chosen odd and $\equiv 29 \pmod{30}$; since no sum of three elements of $A_1$ is $14 \pmod{15}$, three primes from an even block can never reach them, and the deletion rule eliminates the mixed cases.

What would settle it

Choose a moderate starting size and run the block construction for the first few blocks; the claim predicts that each trap integer $x_{2k+1}\equiv 29 \pmod{30}$ has no three-prime representation from $Q$, and that the number of deleted primes is tiny compared with the block size. Finding a representation, or finding deletion counts comparable to the block size, would refute the proof.

Watch

Extended reading notes

Core claim

The central discovery is that the $\tfrac{5}{8}$ density threshold for the three-prime representation theorem is best possible: there exists an infinite set of primes $Q$ with lower relative density exactly $\tfrac{5}{8}$ and upper relative density $1$ for which infinitely many odd integers $2n+1$ are not sums of three elements of $Q$. The proof interleaves blocks of all primes with blocks of primes in the five residue classes $1,2,4,7,13 \pmod{15}$, then deletes from each even block every prime that would take part in a representation of a chosen bad integer $x_{2k+1}\equiv 29 \pmod{30}$. A standard sieve bound, quoted as equation (5), keeps the deleted set sparse enough that the densities are unchanged; the residue-class structure and the deletion rule then block every possible three-prime representation. The paper also answers Problem 66 negatively in both parts with explicit counterexamples modulo $12p$ and $30p$.

Load-bearing premise

The construction rests on a known estimate that each number $x_{2k+1}-p$ has very few splittings into a sum of two primes, uniformly for every prime $p$ up to $x_{2k}$ with one constant that works for all blocks; if that uniformity failed, the deletion step could remove too many primes and the set's lower density would fall below $\tfrac{5}{8}$.

Editorial extensions

If this is right

The strict inequality in the density three-prime theorem cannot be weakened to equality: at lower density exactly $\tfrac{5}{8}$, the conclusion can fail.
Problem 66 is fully resolved: both parts are false as originally stated, while the ternary statement becomes true and sharp when the threshold is $\tfrac{5}{8}$ and $m$ is odd and square-free.
A single residue-class obstruction, $14 \pmod{15}$, can force infinitely many odd integers to be unrepresentable even when the prime set has upper density $1$.
The counterexample sets have positive lower density, so the failure is not a consequence of sparseness but of how the missing primes are distributed.

Reading between the lines

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

Deleting additional sparse primes from the constructed $Q$ preserves the infinitely many unrepresentable odd integers while lowering the lower relative density arbitrarily below $\tfrac{5}{8}$, so the failure phenomenon is not confined to the exact threshold.
The recursive definition sets $x_{2k+2}=e^{e^{x_{2k+1}}}$, so the unrepresentable integers guaranteed by the theorem are of tower-of-exponential size; effective versions would require explicit constants in the sieve bound.
A natural testable extension is whether failure can be forced for almost all odd integers, rather than infinitely many, at some fixed lower density, and what the maximal exceptional set can be at density $\tfrac{5}{8}$.
The alternating-block mechanism should transfer to other small moduli: any subset of reduced residue classes whose ternary sumset misses a class would yield a similar sharpness example if the analogous representation bound holds.

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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

Dear colleague,

The headline: this paper proves the threshold in Shao's density version of ternary Goldbach is sharp. The authors construct a set Q of primes with upper relative density 1, lower relative density 5/8, and infinitely many odd integers that are not sums of three primes from Q. That directly matches Shao's Proposition 2, so the 5/8 boundary is exactly right.

The main theorem is a genuine new result. Yang and Togbé had a similar construction with lower density 1/3; the double-exponential block construction here pushes it to 5/8. The even-modulus counterexamples to Sárközy's Problem 66(a) are also new, and the dyadic recursion is tidy. The paper gives credit where due: Shao's residue set mod 15 is the engine, and the Selberg upper-bound sieve is the only external analytic input. I checked the stress-test concern about equation (5); it is the standard uniform bound on Goldbach representations, and since Q2k is a subset of W2k, the restricted count inherits the upper bound. No problem there.

The soft spots are minor. Fact 1's density preservation argument is compressed; it works because the removed elements lie above x_{2k+1}/sqrt(log x_{2k+1}) and their total count is o(x_{2k+1}/log x_{2k+1}), so Q(x) ~ W(x). But the reader has to fill in the details. In Fact 2, the case split is correct but the exhaustiveness is implicit: because q1<=q2<=q3, any element outside Q2k is from an earlier block and hence <= x_{2k}, so the only possible patterns are the ones listed. The text could say this in one sentence. There are a few typos ("squre-free", "predicition"), and the abstract's "sparse subset" is the wrong word for a set with density 5/8. The note about Helfgott's final version is stale but harmless.

All told, the mathematics holds up. The construction is explicit, the estimates are standard, and the conclusion is sharp. It's a good short paper for additive number theorists. I would send it to a competent referee and expect acceptance after minor revision. Definitely worth a cite.

Referee Report

3 major / 5 minor

Summary. The paper addresses two problems from Sárközy's 2001 collection. For Problem 66, it gives explicit counterexamples showing that the answer is negative for both parts: for part (b), a construction modulo 30p with density 5/8 of the reduced residue classes fails to cover a positive proportion of odd classes, and for part (a), a construction modulo 12p and a self-similar construction modulo 30·2^k fail to cover all even classes. It also quotes Shao's positive result showing that 5/8 is sharp for odd squarefree moduli. The main result, Theorem 2, concerns Problem 67: there exists an infinite set Q of primes with upper relative prime density 1 and lower relative prime density 5/8 such that infinitely many odd integers are not representable as a sum of three elements of Q. This shows that Shao's density threshold 5/8 in Proposition 2 is best possible. The proof constructs Q in alternating intervals: in odd-indexed intervals all primes are taken, in even-indexed intervals only primes in five residue classes modulo 15 (density 5/8), then removes a thin top interval from even blocks (Step 2) and removes all primes that appear in representations of x_{2k+1}-p as sums of two primes from the even block (Step 3). A Selberg sieve estimate bounds the number of removed primes, and a case analysis shows that x_{2k+1} itself is not in Q+Q+Q.

Significance. If correct, Theorem 2 is a substantial and somewhat surprising result: it shows that Shao's density theorem has the sharp threshold 5/8, in the sense that a set of primes with lower relative density exactly 5/8 can fail the ternary representation property. This complements the earlier counterexample of Yang and Togbé, whose lower density was only 1/3. The paper also contributes clean counterexamples for Problem 66 and clarifies the statement of the problem by noting that 'reduced' is necessary. The proof is elementary in structure and relies on a classical Selberg sieve estimate, which is a standard and robust tool. The main strengths are the explicit recursive construction, the uniform sieve bound, and the careful case analysis in Fact 2. The result is likely to be of interest to researchers in additive number theory and prime density problems.

major comments (3)

[Section 3, Step 1, definition of H] The claim that the alternating construction gives d(H)=1 and d(H)=5/8 is plausible but is stated without proof. Since the intervals grow double-exponentially, each new interval contributes a density close to either 1 or 5/8, and previous intervals become negligible; however, the limiting argument should be made explicit, especially because the lower density is defined by a liminf over all x and one must verify that no intermediate value of x dips below 5/8.
[Section 3, Step 3, equation (5)] The Selberg sieve estimate in (5) is the main external input, and its uniform applicability should be justified more carefully: the number N=x_{2k+1}-p varies with p, and the product over prime divisors is bounded by O(log log N) only if one invokes the standard estimate σ(rad(N))/rad(N) ≪ log log N. The authors should state explicitly that this bound is uniform for all N in the interval [x_{2k+1}-x_{2k}, x_{2k+1}] and that the implied constants in (5) are absolute; as written, the transition from the first inequality to the second is a bit terse.
[Section 3, Fact 1, density preservation] The proof of Q(x)∼W(x) should be written out for arbitrary x, not just for x near x_{2k+1}. In particular, one should note that all deleted primes in the k-th even block lie above x_{2k+1}/√log x_{2k+1}, and that the total number of primes deleted in all previous blocks is o(x/log x) for x in the current block; this is true because x_{2k+1}/√log x_{2k+1} vastly exceeds x_{2k}, but the argument deserves a sentence to avoid leaving the impression that only the locally deleted primes are controlled.

minor comments (5)

[Abstract and Section 1, Problem 66 statement] The word 'reduced' is missing in the statement of Problem 66(a) and (b); the authors note this at the start of Section 2 and provide a counterexample for non-reduced sets, but the main text of Section 1 should perhaps flag this immediately, since the problem as quoted is literally false.
[Section 2, construction modulo 30p] In the construction A={30k+1,30k+7,30k+13,30k+17,30k+19: 1≤30k≤m}, the notation is slightly ambiguous: the variable k should be an integer, and the condition should probably be 0≤30k<m or 1≤30k≤m with a clear convention. Also, the assertion |A|≥m/6−5 should be derived explicitly: there are 5 residue classes, each contributing about m/30 elements, with at most one multiple of p removed per class.
[Section 2, modulo 12 construction] The sentence 'Similar discussions as above lead to |A|≥2φ(m)/3 as well as that no element of {12k+4 (mod m):1≤12k≤m} is in A+A' should be rephrased for clarity; it is missing a verb and the reader must infer that the omitted element classes are exactly the even classes congruent to 4 mod 12.
[Section 3, Step 2, interval notation] In equation (2), the interval [x_{2k+1}-x_{2k}-x_{2k+1}/√log x_{2k+1}, x_{2k+1}] is written as a set difference for W_{2k}; it would be clearer to state explicitly that W_{2k} consists of primes in (x_{2k}, x_{2k+1}] that are not in this interval, since the interval may not be integer-aligned.
[Section 3, Fact 2, Case IV] The statement 'By (3) and (4), x_{2k+1}-q_1 ≠ q_2+q_3' is correct only because q_1 runs over primes p with x_1<p≤x_{2k}; the reader should be reminded that q_1 could be ≤x_1, but q_1 is assumed not in Q_{2k} and q_1≤x_{2k} (since q_1≤q_2≤q_3 and q_1+q_2+q_3=x_{2k+1}), so q_1 is indeed a p in that range. This is implicit in the construction and should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained, uses external classical results (Dirichlet, Selberg sieve) and an external counterexample set, and does not assume its conclusion.

full rationale

The paper's central claim, Theorem 2, is a counterexample construction: it builds Q by taking a set H of primes with densities 1 and 5/8 and then deleting from each interval W_{2k} all primes that would participate in a representation of x_{2k+1} - p as a sum of two elements of W_{2k}. The proof that not too many primes are deleted relies on the standard Selberg upper-bound sieve quoted from Nathanson in equation (5), which is an external, independent result, not a self-citation and not an input equivalent to the target statement. The congruence obstruction (that no 29 mod 30 integer lies in the ternary sumset of A0) comes from Shao's explicitly constructed set A1 = {1,2,4,7,13} mod 15, which is used as an external example and benchmark, not as a premise that assumes Theorem 2. The density statement Fact 1 follows from Dirichlet's theorem in arithmetic progressions and from the explicit estimate (6), while the unrepresentability of x_{2k+1} is proven by the case analysis in Fact 2, which derives a contradiction from the deletion rule (3)-(4); the unrepresentability is not assumed at the outset. No parameter is fitted to data and then renamed a prediction, no uniqueness theorem is imported from the authors' prior work, and no equation reduces to the conclusion by construction. The only external inputs are classical sieve bounds and Dirichlet's theorem, which are independent support. Therefore the derivation chain is not circular, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters are used. The construction depends on standard analytic number theory inputs (PNT, Dirichlet's theorem, Selberg sieve) and on Shao's external results that frame the optimality statement. The set Q is a mathematical construction, not a new postulated entity.

assumptions (5)

standard math Dirichlet's theorem on primes in arithmetic progressions: each reduced residue class modulo 15 contains asymptotically 1/8 of the primes.
Used in Step 1 to assert H has lower density 5/8 and in Section 3 to compute Q(x).
standard math Prime number theorem: pi(x) ~ x / log x.
Used to estimate sizes of intervals and to show removed intervals have o(x/log x) primes in (2) and (7).
standard math Selberg upper bound sieve estimate for the number of representations of an integer as a sum of two primes (Nathanson Theorem 7.2).
Used in (5) to bound t_p, the count of representations of x_{2k+1}-p as w+v with w,v in W_{2k}.
domain assumption Shao's finite example: the residue set A1 = {1,2,4,7,13} modulo 15 satisfies |A1| > phi(15)/2 and 14 mod 15 is not in A1+A1+A1.
Used in Case I of Fact 2 and in Section 3 to show x_{2k+1} = 14 mod 15 cannot be a ternary sum of three elements all in A1 residues.
domain assumption Shao's theorem (Proposition 2): if liminf relative prime density > 5/8 then every large odd integer is a ternary sum from Q.
External benchmark establishing that the constructed counterexample at density 5/8 is optimal. This is not assumed in the proof of Theorem 2 but is needed for the optimality claim.

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Cite this review

Pith. "Pith review of Problems 66 and 67 on sums of residue classes and primes of Andr\'{a}s S\'{a}rk\"{o}zy's collection of unsolved problems." pith.science (2026). https://pith.science/paper/TH4ECKWG

@misc{pith2026250802433,
  author       = {Pith},
  title        = {Pith review of: Problems 66 and 67 on sums of residue classes and primes of Andr\'as S\'ark\"ozy's collection of unsolved problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TH4ECKWG}},
  note         = {Machine review of arXiv:2508.02433}
}

read the original abstract

In this note, we discuss two problems of S\'{a}rk\"{o}zy (2001). In particular, we prove an optimal result on sumsets of sparse subset of primes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Alsetri, Density ternary Goldbach for primes in a fixed residue class

A. Alsetri, Density ternary Goldbach for primes in a fixed residue class. https://arxiv.org/abs/2505.09648

work page arXiv

[2] [2]

Alsetri, X

A. Alsetri, X. Shao, Density versions of the binary Goldbach problem. Acta Arith. 218 (2025), 285–295

work page 2025

[3] [3]

Chipeniuk, M

K. Chipeniuk, M. Hamel, On sums of sets of primes with positive relative density. J. Lond. Math. Soc. (2) 83 (2011), no. 3, 673–690

work page 2011

[4] [4]

Acta Math

Zhen Cui, Hong Ze Li, Bo Qing Xue, On sums of sparse prime subsets. Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 8, 1535–1542

work page 2013

[5] [5]

Davenport, Multiplicative Number Theory, Second edition, Graduate Texts in Mathematics 74 (Springer, New York, 1980)

H. Davenport, Multiplicative Number Theory, Second edition, Graduate Texts in Mathematics 74 (Springer, New York, 1980)

work page 1980

[6] [6]

Effinger, D.R

G.W. Effinger, D.R. Hayes, A complete solution to the polynomial 3-primes problem. Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 363–369

work page 1991

[7] [7]

Gao, On the representation of large even integers as the sum of eight primes from positive density sets

M. Gao, On the representation of large even integers as the sum of eight primes from positive density sets. Bull. Sci. Math. 201 (2025), Paper No. 103597

work page 2025

[8] [8]

Hayes, A Goldbach theorem for polynomials with integral coefficients

D.R. Hayes, A Goldbach theorem for polynomials with integral coefficients. Amer. Math. Monthly 72 (1965), 45–46

work page 1965