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.

2. **(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

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

> \[ > \sum \frac{1}{q_i-n}<\sum_{p \]

> 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.

2. 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 \(

then edited the comment to express scepticism because \(A\) can contain

many primes in the shifted window; the comment points to #855.

3. 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

live page.

2. 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

absolute \(C\) such that

\[ \sum_{i=1}^k\frac1{n-a_i} < C+\sum_{p

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

988a913d8a2929a43238ad9bbc97d02a08aad7fbf39030e6a67106bfca72a2e6

86b0ac34845d58d3a9eff3adca9f8cc0847a369434670e1c7fc3d2190c26c7d9

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

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

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],
)