Erdoes #655 - working report - Erdős Problem a Day

Erdős problem 655 — wave w006

Live-page check: 2026-07-28 UTC.

Claim labels used below:

(a) elementary-rigorous — a complete elementary proof is given here.
(b) rigorous-modulo-named-source — a live-page or primary-source

transcription, with the source identified.

(c) plausible/structural-unverified — context or a search conclusion

not promoted to a theorem.

(d) computational-only — established by the supplied exact program;

finite computation is not used as a substitute for the uniform proof.

0. Mandatory live-page gate

(b) I fetched the live page, its discussion thread, its LaTeX view, and its revision history through the Bright Data browser path. The JavaScript-rendered page says OPEN, 0 claimed proofs, and Currently working on this problem: None. “Interested in collaborating,” “Likes,” “difficult,” “tractable,” and both formalisation-interest markers are also all None. The database box says that the original source is ambiguous. Thus the explicit stop condition is not triggered.

Exact live mathematical statement

The live page begins (24-word verbatim excerpt):

“Let \(x_1,\ldots,x_n\in \mathbb{R}^2\) be such that no circle whose centre is one of the \(x_i\) contains three other points. Are there at least”

It then displays

(1+c)\frac n2

and asks, equivalently and without changing the quantifiers, whether the number of distinct values among

\bigl\{\lVert x_i-x_j\rVert:1\leq i<j\leq n\bigr\}

is at least that quantity for some constant \(c>0\), for every sufficiently large \(n\). The complete original wording is in the linked LaTeX view.

(b) The page lists these known facts:

It calls this a problem of Erdős and Pach.
The hypothesis immediately gives at least \((n-1)/2\) pinned distinct

distances from every point.

Zach Hunter observed that \(n\) equally spaced points on a circle

disprove the literal conjecture.

The page suggests that an extra general-position condition, such as no

three collinear and no four concyclic, was probably intended.

The database flags the original source as ambiguous; it labels a

formalised statement as available and a related OEIS sequence as “Possible.”

(c) The five comments, which the site itself warns are unverified, are:

Przemek Chojecki, 2026-04-22: reports tracing several interpretations back

to older Erdős papers with GPT-5.4 Pro and links an overview whose page 9 contains a summary chart.

Neel Somani, 2026-01-19: asks whether Hunter's counterexample means an

additional intended assumption should be specified.

Nat Sothanaphan, 2026-01-19: proposes general position as one candidate.
Thomas Bloom, 2026-01-19: prefers not to guess, mentions convex position

as another natural interpretation, and suggests leaving the challenge open-ended.

Terence Tao, 2026-01-19: says the database was marked as having an

ambiguous statement.

The page therefore simultaneously records a counterexample to its literal wording and retains the OPEN status because the intended problem is unclear. What follows resolves only the displayed literal formulation and states explicitly what it does not resolve.

1. Exact result

For a finite set \(X\subset\mathbb R^2\), write

\begin{aligned} d_X(p)&=\left|\{\lVert p-q\rVert:q\in X\setminus\{p\}\}\right|,\\ D(X)&=\left|\{\lVert p-q\rVert:p,q\in X,\ p\ne q\}\right|,\\ \Sigma(X)&=\sum_{p\in X}d_X(p). \end{aligned}

Let \(\mathcal A_2(n)\) be the family of \(n\)-point sets satisfying the live-page hypothesis. As usual, and as required by the page's own pinned lower-bound remark, “\(n\) points” means \(n\) distinct points.

Theorem (a). For every \(n\geq1\),

\boxed{ \begin{aligned} \min_{X\in\mathcal A_2(n)}D(X) &=\left\lfloor\frac n2\right\rfloor,\\ \min_{X\in\mathcal A_2(n)}\max_{p\in X}d_X(p) &=\left\lfloor\frac n2\right\rfloor,\\ \min_{X\in\mathcal A_2(n)}\Sigma(X) &=n\left\lfloor\frac n2\right\rfloor. \end{aligned}} \tag{1}

For \(n\geq3\), all three equalities are attained simultaneously by the regular \(n\)-gon.

In particular, the live-page conjecture is false for every proposed constant \(c>0\), not merely for a sparse sequence of \(n\).

2. Proof from first principles

2.1 Lower bound

(a) Fix \(p\in X\in\mathcal A_2(n)\). Points at the same positive distance \(r\) from \(p\) lie on the circle of radius \(r\) centred at \(p\). The hypothesis says that each such distance class contains at most two points. The \(n-1\) points of \(X\setminus\{p\}\) therefore require at least

\left\lceil\frac{n-1}{2}\right\rceil =\left\lfloor\frac n2\right\rfloor \tag{2}

classes. Hence \(d_X(p)\geq\lfloor n/2\rfloor\) for every \(p\). Every pinned distance is also a global distance, so

D(X)\geq\left\lfloor\frac n2\right\rfloor, \qquad \max_p d_X(p)\geq\left\lfloor\frac n2\right\rfloor, \qquad \Sigma(X)\geq n\left\lfloor\frac n2\right\rfloor. \tag{3}

2.2 Uniform construction

(a) Identify \(\mathbb R^2\) with \(\mathbb C\), put \(\zeta=e^{2\pi i/n}\), and take

X_n=\{1,\zeta,\zeta^2,\ldots,\zeta^{n-1}\}. \tag{4}

For two vertices whose cyclic separation is \(k\),

\begin{aligned} \lvert \zeta^i-\zeta^{i+k}\rvert^2 &=2-\zeta^k-\zeta^{-k}\\ &=4\sin^2\!\left(\frac{\pi k}{n}\right). \end{aligned} \tag{5}

The separations \(k\) and \(n-k\) give the same chord. Conversely, on \(1\leq k\leq\lfloor n/2\rfloor\), the argument \(\pi k/n\) lies in \((0,\pi/2]\), where sine is strictly increasing. Thus there are no other equalities.

For a fixed vertex and each \(k<n/2\), exactly two vertices occur at that distance, one in each cyclic direction. If \(n\) is even, the antipodal distance \(k=n/2\) occurs once. Therefore every circle centred at a vertex contains at most two other vertices, so \(X_n\in\mathcal A_2(n)\), and

d_{X_n}(p)=D(X_n)=\left\lfloor\frac n2\right\rfloor \quad\text{for every }p\in X_n. \tag{6}

Summing (6) gives the third equality in (1). The cases \(n=1,2\) are immediate.

The complete pinned multiplicity profile is:

| parity of \(n\) | distinct pinned distances | multiplicities | |---|---:|---| | \(n\) odd | \((n-1)/2\) | every class has size \(2\) | | \(n\) even | \(n/2\) | one class has size \(1\), the other \(n/2-1\) classes have size \(2\) |

2.3 The requested quantifiers

(a) Given any fixed \(c>0\) and any \(n\geq1\), the construction obeys

D(X_n)=\left\lfloor\frac n2\right\rfloor \leq\frac n2 <(1+c)\frac n2. \tag{7}

It is consequently a counterexample for every \(n\), so no choice of \(c>0\) and threshold \(n_0\) can make the displayed live-page assertion true.

3. Scope: what is and is not settled

(a) A regular \(n\)-gon is in convex position and has no three collinear vertices: a line meets its circumcircle in at most two points. Consequently, adding only convex position or only “no three collinear” does not repair the literal statement. Requiring no four concyclic points does exclude this construction.

(a) Equation (1) completely settles the literal local-\(\mathcal A_2\) global-distance question. It also gives the exact pinned and sum minima under that same literal hypothesis.

(b) It does not settle the historically attested question in which one simultaneously requires no four concyclic points and asks for a pinned improvement. That different problem is what the 1988 primary source states. No claim about its resolution is made here. Likewise, the page has not selected one intended repair, so this report does not silently choose one and call it Problem 655.

4. Primary-source and current-state search

4.1 Sources actually verified

(b) P. Erdős, “Some combinatorial and metric problems in geometry” (1987), pp. 167–177, SHA-256 7637fce3170209f9d767d969d8c7c19b58144717f1a86798a4148aad2952a1f9. On printed page 168, Erdős considers points in general position, defines the maximum pinned distance count, records the trivial \((n-1)/3\) bound, and asks for a fixed improvement over \(n/3\). He then asks whether the same might follow from no-four-concyclicity or from the weaker local condition allowing at most three other points on a centred circle. This is an \(n/3\)-scale pinned question, not the literal live-page global question.

(b) P. Erdős, “Some old and new problems in combinatorial geometry” (1988), pp. 32–37, SHA-256 842eed38ccf222bce478ef02265dbae8d93e70289e5ddf89bd8db63f8dfc6f69. Printed page 35 assumes both:

no four points lie on a circle; and
every circle centred at a selected point contains at most two other

selected points.

It asks whether

\max_i d(x_i)>(1+c)\frac n2 \tag{8}

for an absolute \(c>0\), and also suggests the analogous sum bound. The paper explicitly explains that its no-four-concyclic assumption is needed because the regular polygon is otherwise a counterexample. Thus the closest primary antecedent I could inspect differs from the live statement in two important ways: it retains no-four-concyclicity and asks about a maximum pinned count rather than only the global count.

(b) The live page cites [Er97e]. Expanding its bibliography gives P. Erdős, Some of my favourite unsolved problems, Mathematica Japonica 46(3) (1997), 527–537, MR 1487304. The journal issue contents verify the issue, author, title, and pages. I did not locate an accessible primary full-text scan, so I make no exact-content claim about that paper.

4.2 Current linked note and search miss

(c) The most recent comment links the unsigned 11-page note “Erdős Problem #655 and Its Natural Repairs”, dated 2026-04-22, SHA-256 2184dfc3b147fdf08fe365f023921d05d32bd4e30cbcce10069f8d2a946fa32d. It states the same exact regular-polygon theorem (1) and surveys possible repairs. The comment says GPT-5.4 Pro was used to prepare the historical trace; the PDF has no named author in its text or metadata. I therefore treat it as useful secondary context, not as a primary source or as evidence of novelty. The proof above is independent and shorter.

(c) Exact-title, exact-phrase, arXiv, journal, Erdős-archive, and citation searches found the live page, the 1987 and 1988 primary papers, the 1997 journal record, and the 2026 overview, but no later primary paper devoted to the literal formulation. This is a documented search miss, not proof that no other literature exists.

5. Standalone exact verification

The verifier is erdos655_wavew006_reverify.py. Its SHA-256 is 40508c5dde3062b3076b56265aca278e1b29c0b9a0ccdd07dae52e326c2888c7. It uses only the Python standard library for mathematics. Its main engine does not evaluate sines numerically. For a primitive \(n\)-th root \(z\), it computes the cyclotomic polynomial \(\Phi_n\) from scratch and reduces

2-z^k-z^{n-k} \tag{9}

modulo \(\Phi_n\). Since \(\Phi_n\) is the minimal polynomial of \(z\), two such residues agree exactly when the corresponding squared chord lengths agree. The program then:

compares that algebraic partition independently with the cyclic-step

partition \(k\mapsto\min(k,n-k)\);

enumerates every pinned multiset and every global pair;
checks the shell capacity and parity profiles;
optionally checks every triple for non-collinearity by an exact

cyclotomic area determinant; and

optionally downloads, hash-pins, and passage-checks both primary PDFs,

the 2026 note, and the 1997 journal record.

Run:

python runs/erdos655_wavew006_reverify.py \
  --max-n 200 --check-collinearity-through 40

python runs/erdos655_wavew006_reverify.py \
  --max-n 80 --check-sources

(d) The first run checked every \(1\leq n\leq200\) exactly and ended:

n=197: pinned/global counts=98; max shell multiplicity=2; 98 shells of multiplicity 2
n=198: pinned/global counts=99; max shell multiplicity=2; 1 shell of multiplicity 1 and 98 of multiplicity 2
n=199: pinned/global counts=99; max shell multiplicity=2; 99 shells of multiplicity 2
n=200: pinned/global counts=100; max shell multiplicity=2; 1 shell of multiplicity 1 and 99 of multiplicity 2
verified n=1..200; transcript sha256=506a8d1b69c3b60671f34663e2c539532a715d0bc7b90c09b924a7d84987a134
ALL EXACT CHECKS PASSED

The audited source run ended:

Er87b: SHA-256 and 3 source passages verified
Er88: SHA-256 and 3 source passages verified
Ulam 2026 overview (secondary): SHA-256 and 3 source passages verified
Er97e journal contents: existence, issue, title, author, and pages verified; no full-text claim made
verified n=1..80; transcript sha256=98f6a76504b4bb6660d19c3efef1caecb9f9f5c28285293233f740cf33d7be6c
ALL EXACT CHECKS PASSED

These computations are independent checks of the algebra and source transcriptions. The uniform conclusion rests on the elementary proof in Section 2, not on extrapolation from \(n\leq200\).

PROVED: The literal live-page formulation is false and its exact sharp minimum is \(D(X)=\lfloor n/2\rfloor\), attained for every \(n\) by the regular \(n\)-gon; historically intended no-four-concyclic pinned repairs remain outside this result.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger