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Subject and independence

Role: the reviewer, working in a fresh context from the assignment alone. The reviewer took no part in writing the connections page, the reconstruction page or the library cards, and read no earlier review or grade of any of them.

Subject: the source as it stood on 2026-09-28T13:50:48Z (called "the subject commit" below), page wiki/research/erdos_939/connections.md, section "## Problem 939, the remaining parts", lines 406–426: the heading at line 406, the paragraph on the third question at lines 408–418, the paragraph on the second question at lines 420–425; the next H2 is at line 427. The section names no lead page. Charge: refutation of every author-recorded statement in the section.

Reading, at the depth stated:

• The whole connections page at the subject commit: the subject section in full; the preamble, the Conventions and the shared results (Lemma 1, Corollary 1.1, Lemma 2, Theorem 3, Corollary 3.1, Proposition 4 and Corollary 4.1) at statement and hypothesis depth, as interfaces; the other connection sections (r = 4, r = 5, Problems 940, 1107 and 937, the other problems examined, the leads and the sources read) at reading depth, to locate the subject and to check its one cross-reference.
• wiki/research/erdos_939/theorem_1_reconstruction.md at the subject commit, in full, as a premise at its reviewed standing.
• The Statement of Problem 939.
• The library cards of Nitaj (1995), Cohn (1998) and Walsh (2024), and the manuscript card behind Theorem 1.
• The held Walsh PDF in library/diophantine_problems/walsh_2024_question_erdos_powerful_numbers_elliptic_curve/, its text extracted and its page 2 read as an image: Theorem 1.1 and its proof at statement and proof depth.
• docs/verification.md, sections "Grading and claim standing", "Audit checklist — the canonical failure modes", "Whole-claim report" and "Audit checklist"; docs/approach_vetting.md; docs/evidence.md, section "Source fidelity"; docs/math_authoring.md.

Exposures, disclosed and used as premises nowhere:

1. The Problem 939 page keeps its Statement, Status, Source, References and Formalization paragraphs in one block above its first H2 (lines 18–74). Extracting the Statement (lines 18–25) brought the Status paragraph (lines 27–36, standing text) and the Formalization paragraph (lines 56–74) into view. They record the same reading of the second and third questions as the section under review. Every finding below is derived from the reconstruction page, the cards, the held PDF and the page's own conventions, and would be unchanged without this exposure.
2. The four library cards were read in full, beyond their provenance paragraphs (digest, "Bears on" and "Read status" paragraphs included). None carries a review verdict.
3. The opening lines of the section "Durable reports and current standing" of docs/verification.md (guidance only).

Not read: the folder's _index.md, everything under its evidence/ (the existing review and grade), the Current assessment and Known results of any problem page, anything among the private working files or outside the repository, and other sections' reviews; no web search was made.

Restatement

Notation as on the page: for $r\ge2$, a positive integer is $r$-powerful when every prime dividing it does so to exponent at least $r$; a family of positive integers is jointly coprime when the greatest common divisor of all its members is $1$; a polynomial family, in the sense of Corollary 3.1, is a pair $a,b\in\mathbb Z[t]$ with $c=a+b$, not all constant, such that for every sufficiently large integer $t$ in some arithmetic progression the values $a(t),b(t),c(t)$ are positive $r$-powerful integers, with $\gcd(a(t),b(t))=1$ for infinitely many such $t$.

• S1 (third question, literature). For $r=3$ the third question of Problem 939, whether there are infinitely many triples of coprime $3$-powerful numbers $a,b,c$ with $a+b=c$, is answered affirmatively in the refereed literature, and the problem page records this.
• S2 (Corollary 3.1 at $r=3$). No polynomial family of coprime $3$-powerful solutions of $a+b=c$ exists: for $r=3$ there are no $a,b\in\mathbb Z[t]$ with the properties above.
• S3 (the constructions are non-polynomial). The constructions of Nitaj (1995), Cohn (1998) and Walsh (2024) of infinitely many coprime $3$-powerful triples are not polynomial families in the sense of S2. In particular Walsh's construction takes multiples $nP$ of a point $P$ of infinite order on the curve $Y^2=X^3-432p^2$, for an odd prime $p$ for which the curve has positive rank, and the heights of the points $nP$ grow quadratically in $n$.
• S4 (limitation). The binomial identity with monomial substitution, the mechanism of the folder, contributes nothing to the third question at $r=3$.
• S5 (second question at $r\ge6$). For each fixed $r\ge6$, the second question of Problem 939, whether there are at most finitely many sums of $r-2$ coprime $r$-powerful numbers that are themselves $r$-powerful, is answered in the negative by Theorem 1 of the reconstruction page, with "coprime" read as jointly coprime.
• S6 (count of the family). For each fixed $r\ge6$ there are constants $c_r,C_r>0$ such that for all large $X$ the number of tuples with sum at most $X$ produced by the two-parameter family of Corollary 1.1 lies between $c_rX^{2/r^2}/\log\log X$ and $C_rX^{2/r^2}$. The section's sentence attaches this count to "the family it produces"; F2 treats the antecedent.
• S7 (trivial upper bound and comparison). For each fixed $r\ge3$ the number of $(r-2)$-tuples of $r$-powerful positive integers with sum at most $X$ is $O_r(X^{(r-2)/r})$, and since $2/r^2<(r-2)/r$ the family of S6 is a vanishing proportion of that bound.
• S8 (cross-reference). The Problem 940 section of the same page records the gap between the count of S6 and the shifted-target bound of the Problem 940 folder. This is a pointer, not a mathematical claim.

Checklist

• Quantifiers and scope. Pass with one qualifier. Every statement is for a fixed $r$, matching Theorem 1 ("for each fixed $r$ separately") and Corollary 1.1 (constants depending only on $r$, $X$ large depending on $r$). The only scope word the section leaves implicit is the coprimality convention in S5: the site's "coprime" is read as joint, as the page's Conventions and its $r=4$ section state, and Theorem 1's summands are not pairwise coprime (F1).
• Circularity. Pass. S2 rests on Corollary 3.1 (Mason–Stothers), S5 on Theorem 1 (a binomial identity and valuations); neither assumes anything about Problem 939. S3's consistency claim is a consequence of S2, not an input to it.
• Model and convention changes. Pass with the same qualifier. "Polynomial family" is used in exactly the sense Corollary 3.1 defines, and the page's $r=4$ section spells that sense out. Joint versus pairwise coprimality changes the truth of S5 (F1). "Heights" in S3 may be read as canonical or as naive logarithmic heights; both grow quadratically in the multiplier.
• Finite and statistical overreach. Pass. No finite computation or heuristic is used as a proof anywhere in the section.
• Uniformity. Pass. The constants in S6 and S7 depend on $r$ only, and the section's "up to constants" sits in a paragraph fixed at one $r$. The exponent comparison in S7 holds for every $r\ge3$.
• Extremal conclusions. Pass. "Far below" is an exponent comparison, $2/r^2$ against $(r-2)/r$, checked exactly: $2/r^2<(r-2)/r$ is $2<r(r-2)$, true for $r\ge3$. No sharpness is claimed.
• Consequences and composition. Pass with two notes. S5 composes Theorem 1 (hypothesis $r\ge6$; conclusion: infinitely many tuples of $r-2$ positive, pairwise distinct, jointly coprime $r$-powerful integers with $r$-powerful sum) with the site question; the composition is exact under the joint reading. S2 is the "in particular" clause of Corollary 3.1 at $r=3$. S6 is correct for Corollary 1.1's family and misattributed if its pronoun is read as Theorem 1 (F2).
• Computation. Inapplicable to the section, which runs no computation. The reviewer's own checks (the exponent comparison, the joint-but-not-pairwise witness at $r=6$, the count of Theorem 1's own family) are exact integer arithmetic and are recorded under Weakest steps and Findings.
• Reproduction. Inapplicable. The section states no rerun command and cites no evidence run.
• Source and verdict fidelity. Pass for Walsh (2024): the curve $Y^2=X^3-432p^2$, the hypothesis "odd prime $p$ with positive rank" and the multiples $(3pk)P$ of a generator $P$ of infinite order are exactly Theorem 1.1 on page 2 of the held PDF. The quadratic growth of heights is a gloss, not a sentence of the paper; it is a standard property of the multiples of a non-torsion point. For Nitaj (1995) and Cohn (1998) no text is held; their characterization as non-polynomial is entailed by S2 and the cards' recorded results, not read from the sources (F3). "Refereed literature": Bull. London Math. Soc. 27 (1995) and Math. Comp. 67 (1998), as the cards record.

Weakest steps

W1: the transfer behind S5. Theorem 1 gives, for each $r\ge6$, infinitely many tuples $(a_1,\dots,a_{r-2},N)$ with $a_1+\cdots+a_{r-2}=N$, all entries positive, pairwise distinct and $r$-powerful, and $\gcd(a_1,\dots,a_{r-2})=1$. The site asks whether there are at most finitely many sums of $r-2$ coprime $r$-powerful numbers that are $r$-powerful. Under the joint reading each tuple of Theorem 1 is such a sum, and distinct tuples have distinct $N$ (the reconstruction's "Infinitely many tuples" step), so the answer is no. Under a pairwise reading the tuples do not qualify: at $r=6$, with $X=q^6$ and $Y=30^6$, the summands are $(X-Y)^6$, $12X^5Y$, $40X^3Y^3$ and $12XY^5$, and $X^3Y$ divides both $12X^5Y$ and $40X^3Y^3$ (with $q=31$ the greatest common divisor of these two is exactly $12X^3Y$), while the joint gcd is $1$. In general every summand other than $(X-Y)^r$ is divisible by $XY$, so pairwise coprimality fails for every $r\ge6$. The transfer is therefore valid exactly under the reading the page declares.

W2: the characterization of Walsh's construction in S3. Theorem 1.1 of the held paper: for an odd prime $p$ such that $E:Y^2=X^3-432p^2$ has positive rank, and $P$ a generator of infinite order, the multiples $(3pk)P$, $k\ge1$, yield pairwise coprime integer solutions of $x^3+y^3=p^4z^3$; the triple $(x^3,y^3,p^4z^3)$ is then a coprime $3$-powerful solution of $a+b=c$. For a non-torsion point the canonical height satisfies $\hat h(nP)=n^2\hat h(P)$ with $\hat h(P)>0$, and the naive logarithmic height differs from it by a constant depending only on $E$; so the heights of the multiples grow quadratically in the multiplier, whether the multiplier is $k$ or $3pk$. A one-parameter polynomial family, listed along its progression, has $n$-th member of logarithmic size $O(\log n)$, so this growth is incompatible with a polynomial family. Independently of the growth, S2 already forces every construction of infinitely many coprime $3$-powerful triples to be non-polynomial in the corollary's sense, which is what makes S3 true for the two sources whose texts are not held.

W3: the counts in S6 and S7. Corollary 1.1's interface was confirmed: it counts ordered tuples from pairs $(m,B)$ with $B$ a positive multiple of $B_0$ and $m>B$ coprime to $B$, with $N=(m^r+B^r)^r\le X$; its bounds are $c_rX^{2/r^2}/\log\log X$ and $C_rX^{2/r^2}$ for large $X$. Theorem 1's own family, $B=B_0$ fixed and $m=q$ a prime above $B_0$, is smaller: $N\le X$ forces $q\le X^{1/r^2}(1+o(1))$, so its count is asymptotic to $r^2X^{1/r^2}/\log X$ by the prime number theorem, and Chebyshev's bounds give the same order. At $r=6$ and $X=10^{216}$ the exact count is $78{,}488$ against a lower-bound scale $X^{2/36}/\log\log X\approx1.6\times10^{11}$. For S7: every $r$-powerful $n$ can be written $n=A^r\prod_{j=1}^{r-1}B_j^{r+j}$ with each $B_j$ squarefree (for each prime $p$ write $v_p(n)=rm+j$ with $m\ge1$ and $0\le j\le r-1$; put $p^m$ into $A$ when $j=0$, otherwise $p^{m-1}$ into $A$ and $p$ into $B_j$), so the number of $r$-powerful $n\le X$ is at most $X^{1/r}\prod_{j=1}^{r-1}\zeta(1+j/r)$; an $(r-2)$-tuple with sum at most $X$ has every entry at most $X$, which gives $O_r(X^{(r-2)/r})$ tuples. Both counts compose with the exponent comparison of S7 exactly as the section says.

Strongest attack

• S1. Attack: find that a cited construction is unrefereed or answers a different question. The cards record Nitaj (1995) in Bull. London Math. Soc. 27, no. 4, 317–318 and Cohn (1998) in Math. Comp. 67, no. 221, 439–440, both refereed venues, both for the exact third question. Walsh (2024) is an arXiv preprint, but the section does not call it refereed. The attack fails. Limitation: the clause "as the problem page records" concerns text outside the allowed read set; it was seen only through the disclosed exposure and is not relied on.
• S2. Attack: exhibit a polynomial family at $r=3$, or a family outside the corollary's hypotheses that the sentence would still cover. Any $a,b,c\in\mathbb C[t]$, pairwise coprime, not all constant, with every root of multiplicity at least $3$ and $a+b=c$, gives $n_0(abc)\le D$ with $D=\max\deg$, against Mason–Stothers $D\le n_0(abc)-1$; so no such polynomials exist, and the arithmetic hypotheses transfer to the polynomial ones through Corollary 3.1's proof (its interface: values $3$-powerful and positive for all large $t$ in a progression, coprime infinitely often). Integer-valued polynomials with rational coefficients reduce to $\mathbb Z[s]$ on a subprogression, and a several-parameter identity restricts to a line, so widening the sense of "polynomial family" does not produce a counterexample. The attack fails.
• S3. Attack: find a discrepancy with the held text (a different curve, a different set of multiples, a different growth). Theorem 1.1 uses exactly $Y^2=X^3-432p^2$ and the multiples $(3pk)P$ of a generator; the heights of $nP$ grow like $n^2\hat h(P)$. For Nitaj and Cohn no text is held, so the only available attack is the logical one, and S2 closes it. The attack fails; the fidelity limitation is F3.
• S4. Attack: extract a two-summand identity from the mechanism at $r=3$. The binomial identity at $r=3$ reads $(X+Y)^3=(X-Y)^3+6X^2Y+2Y^3$, three summands; merging the last two gives $2Y(3X^2+Y^2)$, which is $3$-powerful at $(m^3,B^3)$ only under a Diophantine condition on $3m^6+B^6$ that holds on no arithmetic progression, and S2 excludes every polynomial family. The attack fails.
• S5. Attack: read "coprime" as pairwise. Then Theorem 1's tuples do not qualify (the W1 witness), and the sentence would be false. Under the page's declared joint reading the attack fails. Attack on the range: Theorem 1 requires $r\ge6$ and the section claims exactly $r\ge6$. The attack fails; the missing qualifier is F1.
• S6. Attack: read "the family it produces" as Theorem 1's own family. Its count is of order $X^{1/r^2}/\log X$, below the stated lower bound for all large $X$ (W3). Under the reading "the family of Corollary 1.1" the bounds are the corollary's own and the attack fails. The ambiguity is F2.
• S7. Attack: find that $X^{(r-2)/r}$ is not an upper bound, or that the exponents are misordered. The bound follows from the $O_r(X^{1/r})$ count of $r$-powerful numbers (W3), and $2/r^2<(r-2)/r$ for $r\ge3$. The attack fails.
• S8. Not a claim; the pointer resolves. The Problem 940 section quotes $0.0556$ against $0.6560$ at $r=6$, which the reviewer recomputed from $2/36$ and $4/6-5/468$.

Premises

• Corollary 3.1 of the connections page (shared result, another reviewer's proof). Interface used: $r\ge2$, $a,b\in\mathbb Z[t]$, $c=a+b$, not all constant, values positive and $r$-powerful for all sufficiently large $t$ in some arithmetic progression, $\gcd(a(t),b(t))=1$ for infinitely many such $t$; conclusion $r=2$ with all three nonconstant, hence no family for $r\ge3$. Consumed at $r=3$ only.
• Theorem 1 of the Theorem 1 reconstruction (premise at its reviewed standing). Interface used: for each fixed $r\ge6$, infinitely many tuples of $r-2$ positive, pairwise distinct, jointly coprime $r$-powerful integers with $r$-powerful sum; its proof's family is $X=q^r$ and $Y=B^r$ with $B$ the product of the coefficient primes and $q>B$ prime; every summand other than $(X-Y)^r$ is divisible by $XY$.
• Corollary 1.1 of the connections page (shared result). Interface used: $r\ge6$; pairs $(m,B)$ with $B$ a positive multiple of $B_0$ and $m>B$ coprime to $B$; distinct pairs give distinct tuples; count with $N\le X$ between $c_rX^{2/r^2}/\log\log X$ and $C_rX^{2/r^2}$ for large $X$.
• Walsh (2024), held PDF, Theorem 1.1 with its proof (pp. 1–3), read at statement and proof depth from the extracted text and the page-2 image. Interface used: odd prime $p$, $E:Y^2=X^3-432p^2$ of positive rank, generator $P$, multiples $(3pk)P$, pairwise coprime solutions of $x^3+y^3=p^4z^3$.
• Nitaj (1995) and Cohn (1998): texts not held; their results (infinitely many coprime $3$-powerful triples with $a+b=c$; in Cohn's construction none a cube) consumed through the cards' recorded statements only. Their methods were not read.
• Problem 939, Statement only: the three questions restated above.
• Standard facts supplied by the reviewer: the Néron–Tate height is a quadratic form with $\hat h(nP)=n^2\hat h(P)$ and $\hat h(P)>0$ for non-torsion $P$, at bounded distance from the naive height; the $O_r(X^{1/r})$ count of $r$-powerful numbers, derived in W3; Chebyshev's bounds for the prime counting function.

Findings

• F1 (suggested). Location: line 420, "The second question at $r\ge6$. Settled negatively by Theorem 1". Defect: the sentence carries no coprimality convention, and its truth depends on one. Witness: at $r=6$ the summands $12X^5Y$ and $40X^3Y^3$ of the Theorem 1 tuple share the factor $X^3Y$; in general all summands but $(X-Y)^r$ share $XY$, so under a pairwise reading of the site's "coprime" Theorem 1 answers nothing. The page declares the joint reading elsewhere (Conventions; the $r=4$ section), and the reconstruction states joint coprimality, so the sentence is correct as the page reads the question. "Settled" is also a status word on a page whose preamble changes no status. Proposed replacement: "Answered in the negative, for each $r\ge6$, by Theorem 1, under the joint-coprimality reading of the question that this page adopts; the summands of Theorem 1 are not pairwise coprime, since all but $(X-Y)^r$ are divisible by $XY$."
• F2 (suggested). Location: lines 421–423, "Corollary 1.1 records the size of the family it produces, between $X^{2/r^2}/\log\log X$ and $X^{2/r^2}$ tuples with sum at most $X$, up to constants". Defect: the pronoun "it" most naturally refers to Theorem 1, whose proof's family ($B=B_0$ fixed, $m=q$ prime) has about $r^2X^{1/r^2}/\log X$ members with sum at most $X$, not the stated order. Witness: at $r=6$ and $X=10^{216}$ the primes $30<q\le10^6$ give $78{,}488$ tuples, while $X^{2/36}/\log\log X\approx1.6\times10^{11}$. The bounds are those of Corollary 1.1's own two-parameter family, which contains Theorem 1's. Proposed replacement: "Corollary 1.1 records the size of the two-parameter family that extends it: between $X^{2/r^2}/\log\log X$ and $X^{2/r^2}$ tuples with sum at most $X$, with constants depending on $r$; Theorem 1's one-parameter family alone has order $X^{1/r^2}/\log X$."
• F3 (note). Location: lines 410–417, "the constructions of Nitaj, Cohn and Walsh are non-polynomial (Walsh's runs through ...)". The characterization is checked against the held text for Walsh only; the Nitaj and Cohn texts are not held, as their cards record. The sentence is true because Corollary 3.1 forces every construction of infinitely many coprime $3$-powerful triples to be non-polynomial in its sense, but the reader is not told that this, and not a reading of the two papers, is the warrant. Proposed replacement: after "are non-polynomial", add "as Corollary 3.1 requires of any such construction; only Walsh's text is held, and his runs through ...".
• F4 (note). Location: the section heading and its two paragraphs. The first question at $r\ge6$ (can such a sum ever be $r$-powerful?) is also answered, affirmatively, by Theorem 1, and the section titled "the remaining parts" does not mention it; the page's $r=4$ and $r=5$ sections cover the first question only at those exponents. No statement is wrong; one sentence would make the coverage complete.
• F5 (note). Location: lines 423–424, "the trivial upper bound $X^{(r-2)/r}$". The bound rests on the $O_r(X^{1/r})$ count of $r$-powerful numbers, which the page nowhere derives or cites. The three-line derivation in W3, or a citation of the classical asymptotic, would make "trivial" self-contained.

Verdict

• S1 (third question answered in the refereed literature): refutation-failed.
• S2 (no polynomial family of coprime $3$-powerful $a+b=c$): refutation-failed.
• S3 (the three constructions are non-polynomial; Walsh's through multiples of a point on $Y^2=X^3-432p^2$ with quadratically growing heights): refutation-failed.
• S4 (the mechanism adds nothing at $r=3$): refutation-failed.
• S5 (second question answered in the negative for each $r\ge6$ by Theorem 1, coprimality joint): refutation-failed.
• S6 (Corollary 1.1's family has between $X^{2/r^2}/\log\log X$ and $X^{2/r^2}$ tuples with sum at most $X$, up to constants depending on $r$): refutation-failed as an interface use of Corollary 1.1; the sentence's pronoun is F2.
• S7 (trivial bound $X^{(r-2)/r}$ and the exponent comparison): refutation-failed.
• S8 (cross-reference to the Problem 940 section): not a claim.

No statement is refuted-as-stated and no proof in the section is defective; the count of required corrections is zero. Limitations: the proofs of Corollary 1.1, Corollary 3.1 and Theorem 1 were not re-verified here beyond their interfaces, since they belong to other reviewers or stand at their reviewed standing; the Nitaj and Cohn texts are not held, so their methods were not checked; the clause "as the problem page records" lies outside the allowed read set and is neither confirmed nor denied by this review; the quadratic growth of heights in S3 is verified as standard mathematics, not as a sentence of the Walsh paper. This focused review assigns no tier and changes no status.