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Subject and independence

Role. The reviewer is an independent reviewer in a fresh context, given only the commissioning assignment, who took no part in writing the page, its shared results, the reconstruction page, or any lead folder, and who read no existing review or grade of this folder. The charge is refutation of the author-recorded mathematics of one section.

Subject. Page wiki/research/erdos_939/connections.md as it stood on 2026-09-28T13:50:48Z, section "## Other powerful-number problems examined", lines 517–560 (the H2 at line 517, content at lines 519–559, the next H2 "## Leads created" at line 561). The section names no lead page; the "Leads created" section of the same page attaches no lead folder to it.

Reading. Read in full: the subject section; the page's opening paragraph and its "Conventions" paragraph (lines 17–56) and the whole "## Shared results" section (lines 58–302: Lemma 1 with Corollary 1.1, Lemma 2, Theorem 3 with Corollary 3.1, Proposition 4 with Corollary 4.1) as inputs whose interfaces were confirmed at each use and whose proofs another reviewer checks; the reconstruction page wiki/research/erdos_939/theorem_1_reconstruction.md in full as a premise at its reviewed standing; the Statement paragraphs only (the text from "Statement." to the line before "Status.") of the thirteen problem pages the section targets, wiki/problems/diophantine_problems/E0364/_index.md, E0365.md, E0366.md, E0938.md, E0943.md, E1081.md, E0941.md, E0942.md, E0935.md, E0137.md, E0936.md, wiki/problems/arithmetic_functions/E0367/_index.md and wiki/problems/factorials_binomials/E1108/_index.md; docs/verification.md (Grading and claim standing, the shared and Erdos-specific Audit checklist, Whole-claim report), docs/approach_vetting.md in full, docs/evidence.md "Source fidelity", and docs/math_authoring.md in full. The section invokes no library source theorem, so no library card, result page, or PDF was read. The other connection sections of the page (Problem 939 at r = 4 and r = 5, the remaining parts, Problems 940, 1107 and 937) were not read.

Exposures. Four, none of them a review, grade, or assessment of the subject: (1) lines 561–585 of the page ("## Leads created" and "## Sources read for this page") were read because the extraction ran to the end of the file; (2) one exploratory read of E0364.md showed, beyond its Statement, the frontmatter status: line, the references block, the "Progress" and "Known Results" sections (both "Not yet compiled"), and the generated library-links block; (3) docs/verification.md lines 639–660, the opening of "Durable reports and current standing", were read past the named sections; (4) a file listing of wiki/research/erdos_939/ showed the names of the two files under evidence/verify/ (a reconstruction review and a grade); neither was opened. The reconstruction page's own "Standing" and "Boundary" paragraphs were read as part of that allowed premise page. The status words for Problems 1081 and 941 that the section quotes were not checked against those pages, since their status text lies outside the read set.

Restatement

The section's conventions are the page's: for $r\ge2$, an integer is $r$-powerful when every prime dividing it does so to exponent at least $r$ ($1$ is $r$-powerful); a nonzero $f\in\mathbb C[t]$ is $r$-powerful when every root in an algebraic closure has multiplicity at least $r$, so constants are $r$-powerful; "powerful" means $2$-powerful. Theorem 3 (shared) reads: for $r\ge2$ and pairwise coprime $r$-powerful $a,b,c\in\mathbb C[t]$, not all constant, with $a+b=c$, one has $r=2$ and none of $a,b,c$ constant; more generally, if $a,b,c$ are $r_a$-, $r_b$-, $r_c$-powerful with all $r_f\ge2$, the sum of $1/r_f$ over the nonconstant members exceeds $1$.

• S0 (opening sentences). The listed pages were surfaced by a search of the problems, research, and theory trees for the objects of the proofs; each is a resemblance without a transfer of the mechanism, and where Theorem 3 says something exact it is stated. A process statement.
• S1 (Problems 364, 365, 366; "Theorem 3 with a constant term"). For every integer $r\ge2$ there is no identity $a+1=c$ in $\mathbb C[t]$ with $a$ nonconstant and $a$, $c$ both $r$-powerful polynomials: such an identity has only two nonconstant terms, which Theorem 3 forbids.
• S2 (Problem 366, "the mixed version"). There is no identity $a+1=c$ in $\mathbb C[t]$ with $a$ nonconstant, $a$ $2$-powerful and $c$ $3$-powerful, because the general clause of Theorem 3 would require $1/2+1/3>1$, and $1/2+1/3<1$.
• S3 (consequence). No polynomial family of consecutive powerful numbers exists. Read under the page's convention for a one-parameter polynomial family (Corollary 3.1 and the paragraph after it): there is no nonconstant $a\in\mathbb Z[t]$ such that $a(t)$ and $a(t)+1$ are both powerful for every sufficiently large integer $t$ in some arithmetic progression; a fortiori no identity of powerful polynomials of that shape; pairs, and hence triples.
• S4 (consistency remark). S3 is consistent with the Pell-equation families of Problem 365, whose members grow exponentially in their index.
• S5 (scoping sentence). Problems 364, 365 and 366 ask for specific pairs and triples, which polynomial identities cannot supply.
• S6 (Problem 938). Consecutiveness of terms in the powerful sequence is a gap condition; identities produce progressions (of squares) and give no control of gaps, so the mechanism has no transfer. A stated obstruction without a witness; no proof is claimed.
• S7 (Problem 943). At $r=2$ the binomial identity of the reconstruction is the Pythagorean parametrization $(m^2+B^2)^2=(m^2-B^2)^2+(2mB)^2$; it yields representations of squares as sums of two squares, in number bounded by a divisor-function quantity, so it neither strengthens nor threatens the conjecture $1_A\ast1_A(n)=n^{o(1)}$ for every $n$.
• S8 (Problems 1081 and 941). Both are settled, the first disproved and the second proved as the section characterizes them, and one-parameter families have no bearing on either.
• S9 (Problems 942, 935, 367, 137, 936, 1108). These concern short intervals, consecutive products, and special sequences, none of which an identity among $r$-powerful values addresses. A scoping judgment.

The section contains no transfer lemma of its own and no conditional deduction; its only exact mathematics is S1, S2, S3, S4 and S7, all consequences of the shared results or of elementary facts.

Checklist

• Quantifiers and scope. S1 is stated without the words "$a$ nonconstant", which the phrase "only two nonconstant terms" presupposes (Finding F2, note; the trivial identity $1+1=2$ among constants is the witness for the literal reading). S3 does not carry the convention under which it holds: for values at infinitely many parameters it is false (Finding F1, suggested; witness $a(t)=8t^2$ on the Pell set). Under the page's convention both are correct. S7's "for every $n$" is the problem's quantifier and is preserved. Pass, with F1 and F2.
• Circularity. None. The section assumes nothing equivalent to its conclusions; Theorem 3 and Corollary 3.1 are external to it. Pass.
• Model and convention changes. The section transports an obstruction about polynomials over $\mathbb C$ to questions about integers. The polynomial-level statements (S1, S2) need no bridge. The values-level statement S3 needs the bridge of Corollary 3.1 (the polynomial-factor proposition of the Problem 940 folder), which the shared section supplies and the section does not cite; the bridge's hypotheses hold with $b=1$ (Weakest step W2). Pass, with F1 asking for the citation.
• Finite and statistical overreach. No finite verification or heuristic average stands in for a proof; the small exact checks in this report are the reviewer's controls, not evidence. Inapplicable.
• Uniformity. No constants, error terms, or family-uniform bounds are asserted. The only quantity, "divisor-function size" in S7, is $n^{o(1)}$ in the problem's own normalization. Inapplicable.
• Extremal conclusions. No infimum, supremum, or sharpness claim is made in the section. Inapplicable.
• Consequences and composition. Each "so" and "hence" was checked separately: "so it is impossible for every $r\ge2$" (S1) follows from Theorem 3's conclusion that no term is constant; "impossible as well" (S2) follows from the general clause; "No polynomial family ... exists" (S3) follows from Corollary 3.1 with $b=1$; "neither strengthens nor threatens" (S7) follows from the family giving lower bounds only and contributing at most $n^{o(1)}$ representations. The composition consumes Theorem 3 and Corollary 3.1 at author-recorded standing. Pass.
• Computation. The two arithmetic facts, the identity $(m^2+B^2)^2-(m^2-B^2)^2=(2mB)^2$ and $1/2+1/3=5/6<1$, were recomputed by hand and in exact integer and rational arithmetic. Pass.
• Reproduction. The section states no rerun command, count, or coverage claim. Inapplicable.
• Source and verdict fidelity. The thirteen problem descriptions were compared with the Statement paragraphs: all match, except that the parenthetical "powerful parts of products of consecutive integers" describes Problem 935 exactly but Problem 367 (a product of powerful parts) and Problem 137 (whether the product is itself powerful) only loosely (Finding F4, note). Problem 365's statement does name Pell equations, as S4 uses. The status words "disproved" (Problem 1081) and "proved" (Problem 941) are characterizations of status text outside the read set and are left to the grader (Verdict, limitations). Pass within the read set.

Weakest steps

W1. Theorem 3 with a constant term (S1 and S2). Re-derived directly from Lemma 2 rather than through Theorem 3's proof. Let $a\in\mathbb C[t]$ be nonconstant of degree $D$, $c=a+1$, so $\deg c=D$ and $\gcd(a,c)=1$ because a common root of $a$ and $a+1$ would be a root of $1$. With $b=1$ the Wronskian of Lemma 2 is $W=ab'-a'b=-a'$, also $W=ac'-a'c=aa'-a'(a+1)=-a'$, nonzero since $a$ is nonconstant. A root of $a$ of multiplicity $\mu$ is a root of $a'$ of multiplicity $\mu-1$, and a root of $c$ of multiplicity $\nu$ is a root of $c'=a'$ of multiplicity $\nu-1$; the roots of $a$ and $c$ are distinct, so the product of $(t-\alpha)^{\mu_\alpha-1}$ over the roots of $ac$ divides $a'$, whence $2D-n_0(ac)\le D-1$, that is, $D\le n_0(ac)-1$. If $a$ is $r_a$-powerful and $c$ is $r_c$-powerful then $n_0(a)\le D/r_a$ and $n_0(c)\le D/r_c$, so

$$D\le\frac{D}{r_a}+\frac{D}{r_c}-1 .$$

For $r_a=r_c=r\ge2$ this reads $D\le2D/r-1\le D-1$, impossible; for $r_a=2$, $r_c=3$ it reads $D\le5D/6-1<D$, impossible. Hence S1 for every $r\ge2$ and S2. The interface of Lemma 2 is met: $a$, $1$, $c$ are nonzero, pairwise coprime, and not all constant. Characteristic zero is essential and is part of the page's convention: over $\mathbb F_p$ the identity $t^p+1=(t+1)^p$ is a $p$-powerful counterexample, which is why the convention must be carried whenever S1 is quoted.

W2. The bridge from polynomials to integer families (S3). The section cites Theorem 3, which speaks about polynomial identities; the claim "no polynomial family of consecutive powerful numbers" speaks about values. The bridge is Corollary 3.1 of the shared section with $b=1$. Its hypotheses with $a\in\mathbb Z[t]$ nonconstant, $b=1$, $c=a+1$: not all constant, yes; for all large $t$ in a progression the values $a(t)$, $1$, $a(t)+1$ are positive $r$-powerful integers, where $1$ is $r$-powerful by the page's convention, so this is exactly the family hypothesis; $\gcd(a(t),1)=1$ for infinitely many $t$, trivially. Its conclusion, that $a$, $b$, $c$ are all nonconstant, contradicts $b=1$, so no such family exists, for any $r\ge2$ and in particular for $r=2$. Triples $a,a+1,a+2$ contain the pair $a,a+1$. Two-parameter families of the Lemma 1 kind reduce to this along lines $V=B$, as the paragraph after Corollary 3.1 records. The corollary's own proof consumes the polynomial-factor proposition of the Problem 940 folder at author-recorded standing; that page is outside this read set and its interface was taken from the corollary's quotation of it. What the bridge does not cover is a polynomial whose values are consecutive powerful numbers only on a sparse parameter set: $a(t)=8t^2$ has $a(t)$ and $a(t)+1=x^2$ both powerful exactly when $(x,t)$ solves $x^2-8t^2=1$, at $t\in\{1,6,35,204,1189,6930,40391,\dots\}$, giving the pairs $(8,9)$, $(288,289)$, $(9800,9801)$, $(332928,332929)$ and so on. These are the Pell families of S4, and S3 is true only because the page's convention runs the parameter over a progression; see Finding F1.

W3. The r = 2 identity and its size (S7). The unsplit binomial identity of the reconstruction, $(X+Y)^r=(X-Y)^r+\sum_{j\ \mathrm{odd}}2\binom rjX^{r-j}Y^j$, at $r=2$ is $(X+Y)^2=(X-Y)^2+4XY$, and $X=m^2$, $Y=B^2$ gives $(m^2+B^2)^2=(m^2-B^2)^2+(2mB)^2$; the $j=3$ split of Theorem 1 does not arise since $3>r$. Each summand is a square, so each is powerful, and the sum is the square $n=(m^2+B^2)^2$. The number of parameter pairs $(m,B)$ producing a given $n$ is at most the number of representations of $\sqrt n=m^2+B^2$ as a sum of two squares, $r_2(\sqrt n)\le4d(\sqrt n)$, and $d(k)=k^{o(1)}$, so the family contributes $n^{o(1)}$ ordered representations of $n$ as a sum of two powerful numbers. Problem 943 asks for an upper bound for all $n$; a construction supplies lower bounds only, and this one is below the conjectured size, so it neither proves nor contradicts the conjecture. S7 is exact as stated.

Strongest attack

• S1. Sought $r$-powerful $a,c\in\mathbb C[t]$, $a$ nonconstant, with $c=a+1$: for squares, $g^2-f^2=1$ factors as $(g-f)(g+f)=1$, forcing constants; for general $r$-powerful shapes such as $f^2g^3$ the abc bookkeeping of W1 leaves no degree $D\ge1$; positive characteristic gives counterexamples but is excluded by the convention. Failed.
• S2. Sought $a$ $2$-powerful and $a+1$ $3$-powerful; W1 with $(r_a,r_c)=(2,3)$ gives $D\le5D/6-1$; the reverse assignment $(3,2)$ is the same inequality. Failed.
• S3. Sought a polynomial family. The Pell witness $a(t)=8t^2$ refutes the strongest reading (infinitely many parameters) but not the page's reading, since $8t^2+1$ is a square on no arithmetic progression of large $t$ (by Corollary 3.1's bridge, a polynomial that is $2$-powerful at all large $t$ in a progression is $2$-powerful as a polynomial, and $8t^2+1$ is squarefree). Failed under the page's convention; Finding F1 asks the sentence to carry that convention.
• S4. The solutions of a Pell equation $x^2-Dy^2=1$ are the powers of the fundamental unit, so the $k$-th pair has size exponential in $k$; consecutive powerful pairs of this kind are $(Dy^2,x^2)$ with $D$ powerful. No attack available beyond confirming the fact.
• S5. Attacked the reasoning rather than the conclusion: Problems 364 and 366 ask whether one triple or pair exists at all, and Problem 365 whether every pair contains a square, so a polynomial family, had one existed, would have supplied infinitely many objects and answered the first two positively and the third negatively; the reason none is supplied is S1, not the specificity of the questions. The conclusion stands; the wording is Finding F3.
• S6. Sought a transfer: an identity $a+c=2b$ among powerful polynomials exists at $r=2$ (three squares in progression), but a three-term progression of consecutive terms of the powerful sequence needs no powerful number strictly between the terms, and no polynomial identity controls the powerful numbers in an interval. No transfer was found, so the obstruction is not defeated; the sentence claims no theorem.
• S7. Sought a family of representations exceeding $n^{o(1)}$ from the identity: the count is bounded by $r_2(\sqrt n)$ (W3). Failed.
• S8. Sought bearing of a one-parameter family on Problem 1081 or 941: a family $n=f(t)$ of degree $d\ge1$ contributes $O(x^{1/d})$ integers up to $x$, negligible against an asymptotic of size $x/\sqrt{\log x}$ and irrelevant to a statement about all large integers. Failed. The status words were not attackable within the read set.
• S9. Sought a use of an identity among $r$-powerful values for one of the six problems. For Problem 935's question whether $Q_2(n(n+1)\cdots(n+\ell))/n^2$ is unbounded, three consecutive integers with large square factors would suffice, but that needs large powerful parts, not powerful values, and S1 shows two consecutive terms cannot both be powerful polynomials; the mechanism of the folder produces only fully powerful values. For Problem 942, a family gives $O(1)$ powerful numbers per interval $[n^2,(n+1)^2)$, against a target of $(\log n)^c$. For Problems 137, 936 and 1108 the objects are not polynomial in a parameter. No transfer was found; the sentence claims no theorem.

Premises

• Theorem 3 and its general clause, shared section of the subject page (lines 186–203), author-recorded, read in full; interface used: $r\ge2$, $a,b,c\in\mathbb C[t]$ pairwise coprime, $r$-powerful in the page's sense with constants $r$-powerful, not all constant, $a+b=c$; characteristic zero. Its proof is another reviewer's subject; W1 re-derives the constant-term case from Lemma 2 independently.
• Lemma 2, the Mason–Stothers inequality, shared section (lines 148–184), classical, proof recorded on the page, read in full; interface: nonzero, pairwise coprime, not all constant.
• Corollary 3.1, shared section (lines 205–238), author-recorded, read in full; interface: $a,b\in\mathbb Z[t]$, $c=a+b$, not all constant, values positive and $r$-powerful for all large $t$ in an arithmetic progression, $\gcd(a(t),b(t))=1$ infinitely often. It consumes the polynomial-factor proposition of wiki/research/erdos_940/finite_core_reduction.md at author-recorded standing; that page was not read (outside the read set), and its interface is taken from the corollary's quotation of it. Used only for the values reading of S3.
• The Theorem 1 reconstruction, wiki/research/erdos_939/theorem_1_reconstruction.md, premise at its reviewed standing (reviewed as it stood on 2026-09-28T05:03:27Z with corrections applied since), read in full; used only for the form of the unsplit binomial identity, which is the binomial theorem.
• The Statement paragraphs of the thirteen problem pages listed under Reading, read in full; used for source fidelity only.
• Elementary inputs: Pell solutions as powers of the fundamental unit; $r_2(k)\le4d(k)$ and $d(k)=k^{o(1)}$; a polynomial family of degree $d$ has $O(x^{1/d})$ values up to $x$.
• No library card, result page, or PDF is invoked by the section, and none was read.

Findings

• F1 (suggested). Location: "No polynomial family of consecutive powerful numbers exists, consistent with the Pell-equation families of Problem 365 growing exponentially." Defect: the sentence does not carry the convention that makes it true, and the result it cites, Theorem 3, covers polynomial identities but not values; the values reading rests on Corollary 3.1 with $b=1$. Under the strongest reading, a nonconstant $a\in\mathbb Z[t]$ with $a(t)$ and $a(t)+1$ both powerful for infinitely many $t$, the sentence is false. Witness: $a(t)=8t^2$ at $t=1,6,35,204,1189,6930,\dots$, the $y$-coordinates of $x^2-8y^2=1$, where $(a(t),a(t)+1)=(8,9),(288,289),(9800,9801),(332928,332929),\dots$ are consecutive powerful pairs. Proposed replacement: "No one-parameter polynomial family of consecutive powerful numbers exists: by Corollary 3.1 with $b=1$, no nonconstant $a\in\mathbb Z[t]$ has $a(t)$ and $a(t)+1$ both powerful for all large $t$ in an arithmetic progression. The Pell-equation families of Problem 365, such as $(8y^2,x^2)$ on $x^2-8y^2=1$, are not of this kind: their parameter runs over the exponentially sparse solution set of a Pell equation."
• F2 (note). Location: "an identity $a+1=c$ with $a,c$ $r$-powerful polynomials has only two nonconstant terms, so it is impossible for every $r\ge2$". Defect: the hypothesis "$a$ nonconstant" is used (it is Theorem 3's "not all constant") but not written; constants are $r$-powerful under the page's convention, so $1+1=2$ is an identity of the literal shape. Proposed replacement: "an identity $a+1=c$ with $a$ nonconstant and $a,c$ $r$-powerful polynomials".
• F3 (note). Location: "The problems ask for specific pairs and triples, which polynomial identities cannot supply." Defect: Problem 365's first question is universal (must every pair contain a square?), and a polynomial family, had one existed, would have supplied infinitely many pairs or triples and answered all three problems; the reason none is supplied is Theorem 3, not the form of the questions. Proposed replacement: "Problems 364 and 366 ask whether a single triple or pair exists and Problem 365 whether every pair contains a square; a polynomial family would have answered all three at once, and Theorem 3 rules every such family out."
• F4 (note). Location: "Problem 935, Problem 367 and Problem 137 (powerful parts of products of consecutive integers)". Defect: the parenthetical describes Problem 935 exactly; Problem 367 bounds the product of the $2$-full parts of consecutive integers, which differs from the $2$-full part of their product ($B_2(2)B_2(3)B_2(4)=4$ against $Q_2(24)=8$), and Problem 137 asks whether the product of $k\ge3$ consecutive integers is itself powerful. Proposed replacement: "(the powerful part of a product of consecutive integers, the product of their powerful parts, and whether such a product is itself powerful)".

Verdict

• S0: not a claim (a process statement; the completeness of the search was not checked and is outside the read set).
• S1: refutation-failed (with F2, a note on the unwritten hypothesis).
• S2: refutation-failed.
• S3: refutation-failed under the page's convention for a one-parameter polynomial family (with F1, suggested); under the strongest reading the sentence would be refuted-as-stated by the Pell witness, which is why the convention should be written into it.
• S4: refutation-failed.
• S5: refutation-failed for its conclusion (with F3, a note on the reasoning).
• S6: not a claim (a stated obstruction with no proof asserted; no transfer was found that would defeat it).
• S7: refutation-failed.
• S8: refutation-failed for the "no bearing" assertion; the status words "disproved" and "proved" were not verified within the read set.
• S9: not a claim (a scoping judgment; no transfer was found).

Limitations. The status characterizations of Problems 1081 and 941 and the completeness of the tree search were not checked, since the status text and the trees lie outside the read set. The Problem 937 section, to which the phrase "identities produce progressions of squares" alludes, was not read; S6 was assessed on its own terms. The proofs of Theorem 3, Lemma 2 and Corollary 3.1 belong to another reviewer; this report re-derived the constant-term case of Theorem 3 from Lemma 2 (W1) and confirmed Corollary 3.1's interface at $b=1$ (W2) but did not audit the polynomial-factor proposition the corollary consumes. No required correction was found. This focused review assigns no tier and changes no status.