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Subject and independence

Role. The author of this report is an independent reviewer working in a fresh context from the commissioning assignment alone. The reviewer took no part in writing the reconstruction page, the library card, its result page or any Lean file, and read no other review of the page. Charge: refutation of the page's statement, deductions, imported theorems, labels and locators, not acceptance.

Frozen subject. The source as it stood on 2026-09-28T05:03:27Z (called "the commit" below), path wiki/research/erdos_939/theorem_1_reconstruction.md, read whole from git at that commit.

Artifact. The one-page PDF price_2026_infinite_r_powerful_sums.pdf in the folder of the manuscript's library card, the card's local typesetting of the downloaded TeX source: one physical page, printed page number 1. Depth: the whole page, twice, first the text layer through layout-preserving extraction, then one page image rendered at 150 dpi, on which every displayed formula (the theorem's tuple display, the definition of $t$, the definition of the $v_i$, the binomial identity and the display numbered (1)) and every inline formula of the proof were read. No canonical conversion sits beside the PDF.

Allowed material actually read. (a) The page, at the frozen commit. (b) The library card above and its result page theorem, at the frozen commit. (c) The provenance paragraph of the Conjectures.io card. (d) The statement paragraph of Problem 939. (e) In docs/verification.md, the shared section "Audit checklist" and the Erdos-specific sections "Whole-claim report" and "Audit checklist"; in docs/evidence.md, the section "Source fidelity"; docs/math_authoring.md whole. (f) File listings, names only, of the research folder and the card folder, and existence checks of the page's wikilink targets at the frozen commit. The page cites no other reconstruction page as an input, so none was read. The Lean file, formal_source.json, the research folder's _index.md, the problem pages E0937, E0940 and E1107, two theory claims and the Problem 940 research folder were not read.

Exposures. Three, disclosed here. The library card and its result page were read whole, so their "Read status", "Mathematics", "Bears on" and "Proof sketch" text reached the reviewer beyond the provenance paragraph and the Statement section. The Problem 939 page has no "Statement" heading, so it was read from its title to the "Current assessment" heading, and its "Status", "Source", "References" and "Formalization" paragraphs reached the reviewer. None of this text warrants any verdict below; the one finding it touches (F1) says so.

Restatement

Fix an integer $r\ge6$. Call a positive integer $n$ $r$-powerful when every prime $p$ dividing $n$ has $p^r\mid n$, so that $1$ and every $r$-th power are $r$-powerful. Call a finite family of positive integers jointly coprime when the greatest common divisor of all its members is $1$; pairwise coprimality is not required. The claim: the set of ordered $(r-1)$-tuples $(a_1,\ldots,a_{r-2},N)$ of positive integers such that

$$a_1+\cdots+a_{r-2}=N ,$$

the $r-1$ entries are pairwise distinct, every entry is $r$-powerful, and the $r-2$ summands $a_1,\ldots,a_{r-2}$ (not $N$) are jointly coprime, is infinite. The quantifier is per exponent: one infinite family for each fixed $r$, with nothing uniform in $r$. Nothing is claimed for $r\le5$. The conventions on $r$-powerful and on joint coprimality are the manuscript's (p. 1: the two sentences before Theorem 1, and the last sentence of the abstract).

Checklist

• Quantifiers and scope. Pass. "For every integer $r\ge6$" and "infinitely many tuples" are carried exactly; the page states that the infinitude is per fixed $r$; both parities of $r$ are covered, the odd-$r$ term $2Y^r$ being treated in the distinctness step; the excluded cases $r=4,5$ are named as excluded; no almost-all or eventual reading appears.
• Circularity. Inapplicable: the proof is an explicit construction and assumes nothing about tuples of the kind it produces.
• Model and convention changes. Pass. The objects are the actual positive integers; the definitions of $r$-powerful and jointly coprime match the manuscript's in content; no relaxed or transformed system is substituted.
• Finite and statistical overreach. Pass. The instances $6\le r\le12$ are labeled "Illustration (not in the source)" and "a sanity check"; the proof is algebraic for every $r\ge6$.
• Uniformity. Pass. The quantities depending on the family parameter are $t$, $C$, $v_\ell$, $P$, $B$ and $q$, each defined for the fixed $r$; no constant is claimed uniform in $r$.
• Extremal conclusions. Inapplicable: no infimum, supremum, attained value or sharpness statement is made.
• Consequences and composition. Pass. Every "hence" was re-derived (see Weakest steps and Strongest attack); the one clause the manuscript owes, the distinctness of the summands, is proved on the page and labeled as supplied; no local claim is consumed.
• Computation. Pass. The page's arithmetic remarks were recomputed here exactly: at $r=6$, $C=40$, $P=\{2,3,5\}$, $B=30$, $q=31$ and coefficients $12,40,12$; and for $6\le r\le12$ with the least prime $q>B$ the sum, the $r$-powerfulness of every term (prime by prime over $P\cup\{q\}$ with cofactor $1$, and by exact $r$-th roots for $(X\pm Y)^r$), the joint gcd and the pairwise distinctness all hold.
• Reproduction. Inapplicable as a rerun: the page retains no evidence and says so. Its claim that those instances were checked was reproduced independently as just stated.
• Source and verdict fidelity. Pass for the theorem, the definitions, the display (1), the proof steps attributed to the manuscript and the locators (Theorem 1, its proof, display (1), physical and printed p. 1). Two prose sentences outside the proof carry characterizations that the page's own material does not warrant (F1, F2).

Weakest steps

1. The split coefficients are distinct and positive ($v_t\ge t$). With $t=\lfloor r/2\rfloor-2$ and $C=r(r-1)(r-2)/3$ one needs $C\ge t(t+1)/2$. Since $1\le t\le r/2$ and $x\mapsto x(x+1)/2$ increases on $x\ge0$, $t(t+1)/2\le(r/2)(r/2+1)/2=r(r+2)/8$. Then $r(r+2)/8\le C$ is, after multiplying by $24/r>0$, $3(r+2)\le8(r-1)(r-2)$, that is $8r^2-27r+10\ge0$. The larger root of $8r^2-27r+10$ is $(27+\sqrt{409})/16<3$, so the inequality holds for every integer $r\ge3$ and in particular for $r\ge6$; the page's route, the value $160-24$ at $r=6$ and monotonicity for $r\ge2$ (the consecutive differences are $16r-19>0$), is also correct. Composition: $v_t\ge t>v_i$ for $i<t$ gives distinct positive $v_\ell$ with sum $C$; the distinctness step later needs the $v_\ell$ distinct, and the positivity of the summands needs them positive.

2. The mixed summands are $r$-powerful. A summand $cX^aY^b$ of (1) has $b\ge1$ and every prime of $c$ in $P$, with $X=q^r$, $Y=B^r$, $B$ the squarefree product of $P$ and $q\notin P$. If a prime $p$ divides $cX^aY^b$ then $p\mid c$, $p=q$ or $p\mid B$. If $p\in P$ then $v_p(cX^aY^b)\ge v_p(Y^b)=rb\ge r$. Otherwise $p=q$, which divides neither $c$ nor $Y$, so $q\mid cX^aY^b$ forces $a\ge1$ and $v_q(cX^aY^b)=ra\ge r$. Every prime of the summand therefore has exponent at least $r$. The two $r$-th powers $(X\pm Y)^r$ are $r$-powerful because $v_p(m^r)=r\,v_p(m)$. Composition: this is the $r$-powerfulness clause of the restatement for all $r-1$ numbers.

3. The summands and the total are pairwise distinct (the supplied step). The $q$-adic valuation of the binomial summand with index $j\in J\setminus\{3\}$ is $r(r-j)$; of each split summand, $r(r-3)$; of $(X-Y)^r$, $0$, because $q\mid X$ and $q\nmid Y$. Distinct $j$ give distinct valuations, and $r(r-j)=r(r-3)$ only for $j=3$, so the only possible coincidences are between two split summands, which are equal only when $v_\ell=v_m$, and, when $r$ is odd, between the two summands of valuation $0$, $(X-Y)^r$ and $2\binom rrY^r=2Y^r$. The latter would give $u^r=2w^r$ for the lowest-terms numerator and denominator of $(X-Y)/Y$; comparing $2$-adic valuations, $r\,v_2(u)=1+r\,v_2(w)$, impossible because $r\ge2$ does not divide $1$. $N$ exceeds every summand as a sum of $r-2\ge4$ positive integers. Composition: this is the distinctness clause; with joint coprimality (a prime dividing all summands divides $X-Y$ and $XY$, hence both $X$ and $Y$, contradicting $\gcd(q^r,B^r)=1$) and the infinitude in $q$ (distinct primes $q<q'$ give $(q^r+Y)^r<(q'^r+Y)^r$), Theorem 1 follows.

Strongest attack

The attack aimed at the exponents that the hypothesis $r\ge6$ barely clears and at the coefficient bookkeeping. First, the count: $|J|+t=r-2$ is forced by the definition of $t$, so the number of summands cannot be wrong, and $t\ge1$ needs exactly $\lfloor r/2\rfloor\ge3$, that is $r\ge6$; at $r=4,5$ the unsplit identity has $3$ and $4$ summands against $r-2=2,3$, as the page says. Second, a coefficient prime escaping $P$: the coefficients of (1) are $2\binom rj$ for $j\in J\setminus\{3\}$ and the $v_\ell$, all of whose primes lie in $P$ by definition; $C=2\binom r3$ itself is not a coefficient of (1), and $v_1=1$ (present when $t\ge2$) contributes no prime, harmlessly. Third, $q\in P$: impossible, since every element of $P$ divides $B<q$. Fourth, a collision among summands: the $q$-adic valuations separate every pair except two split summands (separated by the distinct $v_\ell$) and, for odd $r$, $(X-Y)^r$ against $2Y^r$, both of valuation $0$; that collision would make $2$ a rational $r$-th power, refuted by $2$-adic valuation. Fifth, a $2$-adic defect in the odd-$r$ term $2Y^r=2B^{r^2}$: $2\in P$ always, from $2\binom r1=2r$, so $v_2(2Y^r)=1+r^2\ge r$. Sixth, the joint gcd through a prime of $X-Y$: a prime of all summands divides $XY$ as well, hence both $X$ and $Y$, a contradiction. Every route closed, and the exact recomputation for $6\le r\le12$ found the constants and properties as stated. The attack failed; the argument stands as written on the page.

Premises

• The manuscript (Infinite $r$-Powerful Sums, one page, held as a local typesetting of the downloaded TeX source by the card named above): read whole, text layer and page image. Interface used by the page: Theorem 1 as restated above; the proof's definitions of $J$, $t$, $C$, $v_i$, $P$, $B$, $q$, $X$ and $Y$; display (1). The card records that the snapshot's relationship to the text posted on 24 May 2026 is not known; this review examined the held artifact only.
• Binomial theorem. Standard, no held source; used in the displayed form, with the odd-part identity derived on the page by subtraction.
• Unique factorization in $\mathbb Z$. Standard; used as the existence and additivity of $p$-adic valuations, Euclid's lemma ($p\mid ab$ implies $p\mid a$ or $p\mid b$), $p\mid m^r$ implies $p\mid m$, and the lowest-terms form of a positive rational.
• Infinitude of primes. Standard; used to pick one prime $q>B$ and then infinitely many.
• Local claims consumed. None. Two other claims are mentioned in a relation paragraph only and are not consumed; their content and standing were outside the read set and are not checked here.
• Explicit assumptions. $r$ an integer with $r\ge6$; nothing else.
• Held description of the formal counterpart. The card's "Formal source" paragraph only; the Lean file itself was not read.

Findings

F1.

• Severity: suggested.
• Location: Boundary, "stays open at $r=4$ and $r=5$, and the existence question at $r=4$".
• Defect: a sentence about the standing of Problem 939's questions, whose warrant (literature and catalog searches) lies outside this page and outside the manuscript; the page's Standing paragraph says it changes no status, and a research page is not where status is recorded.
• Witness: the manuscript (p. 1) claims nothing at $r\le5$ and says nothing about openness, and the page's proof establishes nothing at $r\le5$. The excluded Status paragraph of the problem page that reached the reviewer (see Exposures) agrees with the sentence, so no error of fact is asserted; the finding is one of warrant and placement.
• Replacement: "The manuscript claims nothing at $r\le5$, and this page adds nothing there; the standing of the $r=4$ and $r=5$ cases is recorded on the problem page."

F2.

• Severity: suggested.
• Location: Boundary, "The same statement, with positive, IsPowerful, injective summands and joint coprimality as 'no prime divides every summand', is the theorem infinite_rpowerful_sums".
• Defect: "the same statement" is stronger than the held description supports, and the sentence itself disclaims a line-by-line comparison.
• Witness: the card's "Formal source" paragraph names two main theorems, infinite_rpowerful_sums and infinite_rpowerful_sum_tuples, and describes their conclusion as "an infinite set of sums"; Theorem 1 as reconstructed concludes infinitely many tuples, a weaker form. The page's proof does give infinitely many sums $N$, but the statement it reconstructs does not say so.
• Replacement: "A formal counterpart, with positive, IsPowerful, injective summands, joint coprimality as 'no prime divides every summand', and the infinitude stated for the set of sums, is the theorem infinite_rpowerful_sums (with a tuple form infinite_rpowerful_sum_tuples) of the Lean file that the Conjectures.io card records, ...", the rest of the sentence unchanged.

F3.

• Severity: note.
• Location: Source, "shared in the erdosproblems.com forum thread for Problem 939 on 24 May 2026".
• Defect: the page dates the manuscript by the forum post but reads a snapshot accessed on 2026-09-27, and the card states that the snapshot's relationship to the text present on 24 May 2026 is not known; the page does not carry that caveat.
• Witness: the card's "Canonical snapshot" paragraph, last sentence.
• Replacement: after "so the page reference is to that artifact", add "; the card notes that the snapshot's relationship to the text as posted is not known".

F4.

• Severity: note.
• Location: Splitting the cubic coefficient, "which holds for $r\ge6$: at $r=6$ the two sides are $24$ and $160$, and the difference ... increases for $r\ge2$".
• Defect: this verification is the page's, not the manuscript's. The manuscript (p. 1) asserts "$8(r-1)(r-2)\ge3(r+2)$ for $r\ge6$" without proof, and the Standing paragraph names only the distinctness step as supplied. The check is routine and correct (Weakest steps, 1), so nothing is altered; the label is the only point.
• Replacement: add "(the manuscript states the inequality without proof; the check is this page's)".

Verdict

Source fidelity: faithful. The statement's hypotheses (an integer $r\ge6$), conclusion (infinitely many tuples of positive integers summing to $N$, all $r-1$ numbers distinct and $r$-powerful, the $r-2$ summands jointly coprime), quantifiers (per fixed $r$), conventions ($r$-powerful, joint coprimality) and locators (Theorem 1, its proof, display (1), physical and printed p. 1 of the held PDF) match the artifact. No required corrections; two suggested corrections (F1, F2) and two notes (F3, F4), all in prose outside the statement and the proof.

The argument as reconstructed: sound. Every deduction was re-derived; the supplied distinctness step is correct and labeled; the steps attributed to the manuscript are the manuscript's; nothing the manuscript proves is altered or strengthened.

Limitations. The relation paragraph on Problem 940 (the wording of that problem's questions, the content and tier of two other claims, and the remark on the Conjectures.io submission's $r=4$ leg) lies outside the commissioned read set and was not checked. The Lean file was not read, so F2 rests on the card's description. The held PDF is a local typesetting of a snapshot whose relationship to the text as posted is not known. This focused review assigns no tier and changes no status.