Erdős problem #1210 — wave 8k

Access/check date: 2026-07-28 UTC.

Outcome

The problem remains open. This run gives two independently checkable pieces of progress.

1. (a) Elementary-rigorous: an exact radical-support compression formula

for the finite extremal quantity \(M(n)\). It turns the optimization over all pairwise-coprime subsets of \([1,n)\) into a weighted set-packing problem involving only profitable composite replacements of a canonical prime-power baseline.

1. **(d) Computational-only for the finite optimality claims; (a) for

exhaustiveness and every displayed witness:** exact rational computation for every \(2\leq n\leq1000\). In this complete range, \[ M(n)\leq \sum_{p<n}\frac1p+1, \] and the constant \(1\) is sharp, with equality exactly for \(n=2,3,4,6\). For \(16\leq n\leq1000\), the largest defect occurs at \(n=204\) and is \[ \frac{ 79238096535390154478342194618765038722392907207825438 }{ 81727606403062853596766096017766921944452742445065375 } =0.969538935774\ldots. \]

This is not a uniform proof. The exact remaining obstruction is a uniform bound on a weighted positive packing defect; even its prime-only part contains the shifted short-interval problem highlighted on the live page and in the recent literature.

0. Mandatory live-page gate

I used the Bright Data cloud-browser path to read the rendered live pages:

• <https://www.erdosproblems.com/1210>
• <https://www.erdosproblems.com/latex/1210>
• <https://www.erdosproblems.com/forum/discuss/1210>

I did not rely on datacenter curl or on the stale tracker YAML.

Live status and markers

(d: direct browser observation) The rendered page displayed:

• OPEN;
• 0 claimed proofs for this problem;
• Interested in collaborating None;
• Currently working on this problem None;
• Likes this problem hunterbates;
• This problem looks tractable TFBloom, Basile_Beyer_de_Ryke, hunterbates;
• no “difficult”, “formalisable”, or “working on formalising” marker.

The page said it was last edited 08 April 2026. Thus neither mandatory stop condition was present.

Verbatim current statement

The following is copied verbatim from the current LaTeX-source page:

Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$. Is it true that
\[\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p<n}\frac{1}{p}+O(1)?\]

(a, using the original 1980 wording below) The \(O(1)\) is an absolute constant, uniform in \(n\) and in the admissible set \(A\).

Other information printed on the problem page

The page gives references [Er77c,p.64][Er80,p.112], the tag number theory, and this historical note:

In [Er80] he claims he 'did not state [this] quite correctly' in [Er77c]. The problem in [Er77c] which Erdős is presumably referring to states that if \(n<q_1<\cdots<q_k\leq m\) is the set of primes in \((n,m]\) then \[ > \sum \frac{1}{q_i-n}<\sum_{p<m-n}\frac1p+O(1). > \] See also [460] and [950].

The page itself lists no proved bound for #1210.

All three live comments

(d: browser transcription; none is treated as a theorem merely because it is a comment)

1. ebarschkis, 31 May 2026: “This paper claims a few interesting results

on this problem, but no full solution.” “This paper” links to Idriss Olivier Bado's May 2026 ResearchGate preprint checked below.

1. Thomas Bloom, 08 April 2026: reported a GPT suggestion that it should

suffice to prove \[ |A\cap[n-x,n)|\leq \pi(x)+O\!\left(\frac{x}{(\log x)^2}\right), \] initially suggested charging elements having a prime factor \(<x\), and then edited the comment to express scepticism because \(A\) can contain many primes in the shifted window; the comment points to #855.

1. Nat Sothanaphan, 08 April 2026: agreed that the proposed estimate does

not follow from standard sieve theory, since it would imply \[ \pi(x+y)\leq\pi(x)+\pi(y) +O\!\left(\frac{y}{(\log y)^2}\right), \] one of the conjectural estimates discussed at #855.

The forum explicitly warns that comments are unverified. There was no claimed-proof post and no current-worker marker.

1. Primary-source and literature checks

Erdős's original sources

(d: primary-document check) The Rényi archive contains both sources cited on the live page.

1. P. Erdős, Problems and results on combinatorial number theory. III,

Lecture Notes in Mathematics 626 (1977), 43–72, <https://www.renyi.hu/~p_erdos/1977-27.pdf>. On printed page 64 it gives the earlier conjecture for primes \(n<q_i\leq m\), with denominators \(q_i-n\), exactly as summarized on the live page.

1. P. Erdős, A survey of problems in combinatorial number theory, Annals of

Discrete Mathematics 6 (1980), 89–115, <https://www.renyi.hu/~p_erdos/1980-03.pdf>. On printed page 112 it says, “Here is a problem which I did not state quite correctly in III,” takes \(1\leq a_1<\cdots<a_k<n\) pairwise relatively prime, and asks for an absolute \(C\) such that \[ \sum_{i=1}^k\frac1{n-a_i} < C+\sum_{p<n}\frac1p. \]

Thus the modern statement and the intended uniformity agree with the primary 1980 source.

May 2026 dyadic/Farey preprint

(d for bibliographic verification; b modulo the preprint's standard-sieve input) The paper linked by the first comment exists:

I. O. Bado, A Dyadic-Farey Reduction for an Erdős Problem on Pairwise Coprime Sets, May 2026, DOI <https://doi.org/10.13140/RG.2.2.33043.03361>, full record <https://www.researchgate.net/publication/405212040_A_Dyadic-Farey_Reduction_for_an_Erdos_Problem_on_Pairwise_Coprime_Sets>.

It explicitly says it does not claim a full proof. Its stated results are:

• the distance/residue reformulation;
• the order-of-magnitude estimate

\(M(n)\ll\log\log n\);

• a sufficient summable dyadic-block conjecture;
• a composite-removal estimate reducing sufficiently high dyadic ranges to

shifted-prime comparisons;

• the self-rough admissible family.

The paper identifies sharp rough-number counts and shifted primes as the remaining obstruction.

June 2026 average-order preprint not yet on the live page

(d for bibliographic and theorem-statement verification; b modulo the preprint's Buchstab and beta-sieve arguments) Exact-problem searches found a newer, directly relevant primary artifact:

E. Li, An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem, arXiv:2606.17955v1, submitted 16 June 2026, <https://arxiv.org/abs/2606.17955>.

The 36-page preprint defines exactly the same \(M(n)\) and claims:

• Theorem 1.1:

\[ \sum_{n\leq N}M(n) =e^{-\gamma}N\log\log N+O(N); \]

• Theorem 1.4: for almost all \(n\),

\[ M(n)=(e^{-\gamma}+o(1))\log\log n, \] with a quantitative exceptional-set bound, hence the Erdős inequality for a density-one set of \(n\);

• Proposition 5.1: for every \(\varepsilon>0\),

\[ M(n)\leq(2+\varepsilon)\log\log n+C_\varepsilon \quad(n\geq3); \]

• a conditional window-packing implication, finite structural certificates,

and small exact examples at \(n=16,20,30\).

I verified that the arXiv identifier, author, submission date, full text, and these theorem statements exist. I did not independently audit every analytic-sieve step in the preprint, so these are recorded as preprint claims, not as new theorems of this run. They are partial results and do not settle the uniform constant-\(1\) question.

Literature-search miss

(c, deliberately not a theorem) Exact-statement, exact-title, arXiv, and general web searches found the two 2026 preprints above and the original sources, but no primary source claiming a full uniform proof or counterexample. I also found no independent review or citation of the June preprint. This is a report of the searches performed, not a claim that no other relevant work exists.

2. Exact radical-support compression

Define

For every nonempty prime set \(S\) which occurs as a support below \(n\), put

For a prime \(p<n\),

For \(|S|\geq2\), define its replacement gain

Let

and let the exact packing premium be

Compression theorem

(a) Elementary-rigorous. For every integer \(n\geq2\),

Proof

1. The element \(1\) is coprime to everything and has positive weight, so it

occurs in every maximizer. It contributes \(1/(n-1)\).

1. For \(a,b>1\), \((a,b)=1\) exactly when

\(\operatorname{supp}(a)\cap\operatorname{supp}(b)=\varnothing\). Hence an admissible set is a family of disjoint prime supports, with at most one integer per support.

1. For a fixed support \(S\), the weight \(1/(n-a)\) strictly increases with

\(a\). Thus only \(m_n(S)\), the largest integer with that support, can occur in a maximizer.

1. If no chosen nonsingleton support uses \(p\), the singleton

\(u_p(n)\) can be added. Therefore every maximizer is a canonical baseline \[ \{1\}\cup\{u_p(n):p<n\} \] with some disjoint groups of singleton supports replaced by their nonsingleton \(m_n(S)\).

1. Replacing the singleton group \(S\) changes the weight by exactly

\(\Delta_n(S)\). A nonpositive-gain replacement can be undone without affecting any other chosen support. Thus only \(\mathcal C_n\) matters, and maximizing the sum of disjoint positive gains gives (2.1). \(\square\)

Safe dominance used in the computation

(a) Elementary-rigorous. If \(T\subseteq S\) and \(\Delta_n(T)\geq\Delta_n(S)\), candidate \(S\) may be deleted. Any packing using \(S\) remains feasible after replacing it by \(T\), uses no new prime, and does not lose weight. This is the only nontrivial pruning applied before the exhaustive recurrence.

(c) I did not find formula (2.1) stated in the checked sources. This is a search impression, not a priority or novelty claim.

3. Exact analytic reduction and the prime obstruction

For an admissible \(A\), let \(N=n-1\) and

Discrete summation by parts gives the exact identity

Applying the same identity to the distances which are primes gives

(a) This is an exact finite identity; the verifier checks both sides for every computed extremizer and for the prime benchmark.

A uniform bound

for any fixed \(\delta>0\) would make the positive error in (3.1) summable and would prove #1210.

(a) Exact obstruction to treating (3.2) as a routine sieve estimate. Take \(A\) to be all primes below \(n=x+y\) and set \(m=y\). Then (up to harmless endpoint changes)

Consequently (3.2) would imply

For \(\delta=1\), this is precisely the implication identified in the live comments. Thus the proposed window bound already contains a conjectural short-interval prime statement; pairwise coprimality supplies no extra help when \(A\) itself is the prime set.

Formula (2.1) exposes the same issue differently. If \(p>\sqrt n\), then \(u_p(n)=p\), so its baseline contains

(a) Hence even before any profitable composite replacement is considered, a uniform proof must control a shifted-prime harmonic tail. The exact remaining task in the compressed model is

uniformly in \(n\). Standard one-dimensional upper-bound sieve estimates give the right \(\log\log n\) order but, as the checked preprints explain, lose the sharp leading constant.

4. Exact computation for \(2\leq n\leq1000\)

Algorithm and why it is exhaustive

(a) A smallest-prime-factor sieve computes every \(\operatorname{supp}(a)\). Formula (2.1) constructs all positive-gain candidates, and the safe dominance rule removes only provably redundant ones. A candidate is represented by the bit mask of its prime support.

For an active candidate set \(C\), choose any \(i\in C\). Every optimal packing is in exactly one of the following branches:

where \(N[i]\) is \(i\) together with all candidates sharing a prime with it. Disconnected conflict components are additive. Memoizing (4.1) is an exhaustive exact algorithm, not a heuristic or floating-point MILP.

(d) All weights and comparisons use Python Fraction. For every output row the program independently:

• reconstructs a maximizing set \(A\);
• checks every pairwise gcd;
• recomputes its exact reciprocal sum;
• recomputes the prime benchmark and self-rough sum;
• checks the discrete partial-summation identities;
• checks that the self-rough family has disjoint prime supports.

For \(2\leq n\leq40\), a second optimizer ignores the compression entirely: it processes every original integer \(2\leq a<n\) and dynamically stores the best exact weight for every used-prime mask. It agrees with (4.1) in every case. The program also reproduces the exact \(n=16,20,30\) examples in Li's preprint.

Complete finite-range statement

(d for optimality; a for exhaustive recurrence and arithmetic checking). For every \(2\leq n\leq1000\),

Equality occurs exactly at

For \(16\leq n\leq1000\), the exact maximum defect is the fraction displayed in the Outcome and is attained at \(n=204\).

At \(n=204\), a compact description of the maximizing set is:

• start with \(1\) and \(u_p(204)\), the largest power of each prime

\(p<204\);

• replace the ten singleton supports involved in

\[ \{5,37\},\{11,17\},\{3,67\},\{2,101\},\{7,29\} \] by \(185,187,201,202,203\), respectively.

The resulting explicit witness is

(a) Its pairwise coprimality and weight are directly checkable. Its exact weight is

(d) Recurrence (4.1) certifies that no admissible set has larger weight.

Verified checkpoint table

All displayed decimals are rounded from exact Fraction values. Here

and

is the self-rough lower weight used in the June preprint.

The five largest defects in \(16\leq n\leq1000\) are:

The SHA-256 digest of the canonical exact rows (exact \(M,H_P,L\), compact replacement witness, and full witness for every \(2\leq n\leq1000\)) is

988a913d8a2929a43238ad9bbc97d02a08aad7fbf39030e6a67106bfca72a2e6

5. Reproduction and code

The standalone dependency-free verifier is:

runs/erdos1210_wave8k_verify.py

Source SHA-256:

86b0ac34845d58d3a9eff3adca9f8cc0847a369434670e1c7fc3d2190c26c7d9

Run from the repository root:

PYTHONHASHSEED=271828 python runs/erdos1210_wave8k_verify.py \
  --limit 1000 --direct-up-to 40

Environment used: CPython 3.12.3. The final complete run took 47.858 seconds on this VM and printed, in part:

DIRECT_CROSSCHECK=2..40
LITERATURE_EXAMPLES=16,20,30 MATCH (when within range)
EXACT_RANGE=2..1000
MAX_DEFECT=1/1 (1.000000000000) AT_N=[2, 3, 4, 6]
MAX_DEFECT_N_GE_16=79238096535390154478342194618765038722392907207825438/81727606403062853596766096017766921944452742445065375 (0.969538935774) AT_N=204
MAX_DEFECT_N_GE_500=733118969115751048968118684513145263833659469944488351605998795122449980610595163944099861144923070139556123371438381/765533449251710323760663599607882550042388743057861947713564920595729228607783618823470518780923893017414270940583875 (0.957657656674) AT_N=504
SHA256=988a913d8a2929a43238ad9bbc97d02a08aad7fbf39030e6a67106bfca72a2e6
ELAPSED_SECONDS=47.858

Repeated full runs, including one with the displayed hash seed, produced the same exact-row digest.

The core recurrence in the adjacent full source is:

excluded_value, excluded_choice = solve(active & ~(1 << pivot))
included_value, included_choice = solve(active & ~conflict[pivot])
included_value += candidates[pivot].gain
included_choice |= 1 << pivot
if included_value > excluded_value:
    return included_value, included_choice
if included_value < excluded_value:
    return excluded_value, excluded_choice
return min(
    (included_value, included_choice),
    (excluded_value, excluded_choice),
    key=lambda pair: pair[1],
)

The full implementation also includes component splitting, witness reconstruction, the independent uncompressed dynamic program, all arithmetic checks, and the canonical digest.

6. What remains and why the standard machinery stalls

1. Uniformity. (a) No finite computation can prove (3.3) for all

\(n\). The verified \(+1\) bound through \(1000\) is a sharp theorem only for that concrete finite range.

1. Prime-only missing lemma. (a) Any uniform local estimate strong

enough to make (3.1) summable applies to \(A=\{\text{primes}<n\}\) and yields a conjectural short-interval prime-counting inequality. This is an exact implication, not merely an analogy. A successful proof would need either that prime input or a genuinely weighted cancellation mechanism weaker than pointwise window control.

1. Rough-composite missing lemma. (b, modulo the checked preprints)

Standard upper-bound sieves control the number of rough shifted integers with the correct order \(D/\log D\), but not the sharp constant needed after summing harmonic dyadic blocks. Li's preprint reaches \(2+\varepsilon\) pointwise and constant \(e^{-\gamma}\) on average/almost everywhere; neither supplies the uniform \(1+O(1/\log\log n)\)-scale control implicit in #1210.

1. Exact compressed target. (a) In the new finite formula, the missing

uniform statement is exactly (3.3): jointly bound the shifted prime-power baseline defect and the optimum disjoint positive-replacement premium \(P(n)\). Bounding either term crudely by \(O(\log\log n)\) loses precisely the information sought.

1. Computation cost. (d) The full \(n\leq1000\) sweep took about 47

seconds. In exploratory timing, the selected instance \(n=1500\) took about 1.9 seconds, while \(n=2000\) was still running after 60 seconds and was interrupted. The recurrence is exponential and highly instance-dependent. A naive complete sweep through \(2000\) should be budgeted at several core-hours (rough planning range 5–20 core-hours), not a few CPU-minutes; a serious extension should use independently checkable branch certificates or an exact ILP/SAT backend. Such a sweep would still not address uniformity.

Claim ledger

• (a) Elementary-rigorous: compression formula (2.1) and its proof; safe

dominance; exhaustive recurrence (4.1); explicit \(n=204\) witness; partial-summation identity (3.1); implication from a window estimate to the prime-counting estimate; identification of the exact compressed target.

• (b) Rigorous-modulo-named-theorem/source: Erdős's primary statements;

Bado's order-of-magnitude result modulo the standard upper-bound sieve; Li's claimed average, almost-all, and \(2+\varepsilon\) results modulo the preprint's named Buchstab/beta-sieve inputs.

• (c) Plausible/structural-unverified: only the negative literature-search

report, the impression that (2.1) is not in the checked sources, and the diagnosis of what kind of new mechanism is likely needed.

• (d) Computational-only: live browser observations; document and theorem

location checks; exact optimality for \(2\leq n\leq1000\); timings, state counts, digests, and direct-program cross-checks.

PARTIAL: Exact radical-support compression proved and a from-scratch rational verifier certifies the sharp bound \(M(n)\leq\sum_{p<n}1/p+1\) for every \(2\leq n\leq1000\); the uniform problem remains blocked by the shifted-prime/rough-window constant.