A Polynomial-Identity Route to Infinitely Many Solutions at r = 5

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evidence/: Two exploratory scripts searching numerically for four-term identities of fifth powers of binary forms; no entry point, no checks, and no evidence for any claim.

This lead's notes that cite no source are author-recorded research, not claims.

Target and status evidence

The second question of Problem 939 at $r=5$: a $5$-powerful number is a positive integer divisible by the fifth power of each of its prime factors; are there at most finitely many sums of three jointly coprime $5$-powerful numbers that are themselves $5$-powerful? The problem page records this as open: two solutions are known, both recomputed on the problem page, and a forum-reported search found no further solution with sum below $10^{16}$ except possibly ones in which every pair of summands shares a factor. The same question is answered negatively for every $r\ge6$ by the binomial construction reconstructed in the Problem 939 research folder, and its $r=4$ case is excluded from the polynomial-identity route of Corollary 3.1 of the connections page, one-parameter families and polynomially parametrized parameter curves, by the theorem recalled below. This lead concerns $r=5$ only and the exact erdosproblems.com question, with "coprime" read as joint coprimality, as the problem page records erdosproblems.com and the formal-conjectures catalog read it.

Route

The route is the mechanism of the $r\ge6$ construction with the binomial identity replaced by an unknown identity with the right number of terms. Every link is proved on the Problem 939 connections page; this page states the chain and its standing.

Established inputs, at their actual scope.

• The transfer (Lemma 1 of the connections page; author-recorded, with proof). Let $A_1+A_2+A_3=A_4$ be an identity in $\mathbb Z[U,V]$ in which each $A_i$ is an integer constant $c_i$ times a product of polynomials raised to powers at least $5$, $V^5$ divides every $A_i$ with $|c_i|>1$, $A_1=K^5$ with $K(U,0)=\pm U^{d}$ and $K(0,V)=\pm V^{d'}$, and $A_2$ is a monomial. Then for every $B$ divisible by the primes of the $c_i$ and every $m$ coprime to $B$ at which all $A_i(m,B)$ are positive, the three values $A_1(m,B),A_2(m,B),A_3(m,B)$ are jointly coprime $5$-powerful integers with $5$-powerful sum $A_4(m,B)$; positivity holds for all large $m$ when the leading coefficients in $U$ are positive, and then infinitely many distinct sums arise.
• The three-term obstruction (Theorem 3 and Corollary 3.1 there; author-recorded, with proof from the Mason–Stothers inequality). No identity $a+b=c$ of pairwise coprime $r$-powerful polynomials, not all constant, with $r\ge3$ exists, so the three summands of the premise cannot include two proportional polynomials: the identity must have four genuinely distinct terms, and the four-summand Brownawell–Masser bound recalled on the Corvaja–Zannier source page, $\max\deg\le3(n_0-1)$ with $n_0$ the number of distinct linear factors of the product, permits such identities in every degree at least $3$.
• A complex identity in the least possible degree (Proposition 4 and Corollary 4.1 there; author-recorded, with proof). For a primitive cube root of unity $\omega$ and $Q_k=\omega^{2k}U^2+UV-2\omega^kV^2$, one has $Q_0^5+Q_1^5+Q_2^5=(3UV)^5$, a four-term identity of fifth powers of quadratic forms over $\mathbb Q(\omega)$ with no vanishing proper subsum and no common factor. No projective linear change of variables and rescaling makes all four forms real, so it yields no identity with four rational terms.

Provisional premise. (P) There is an identity $A_1+A_2+A_3=A_4$ in $\mathbb Z[U,V]$ satisfying the hypotheses of the transfer with $r=5$, with positive leading coefficients in $U$ and $A_4(U,B)$ nonconstant for some admissible $B$.

Deduction. Under (P) the transfer produces infinitely many jointly coprime triples of $5$-powerful integers with $5$-powerful sum, answering the second question of Problem 939 at $r=5$ in the negative. Whether the summands are also pairwise distinct, which the formal-conjectures catalog's finite-set formulation requires and the erdosproblems.com question does not, follows for all but finitely many $m$ whenever $A_i(U,B)\ne A_j(U,B)$ as polynomials in $U$.

What is known about the premise

Over $\mathbb C$ a four-term identity of $5$-powerful forms with one fifth-power term and one monomial term exists: the complex identity above has degree $10$, the least possible degree, since degree $5$ would need four fifth powers of linear forms with zero sum, which the Vandermonde determinant of their roots excludes except with vanishing subsums; and it is the only nondegenerate identity of fifth powers of quadratic forms that the search below found. Over $\mathbb R$, and hence over $\mathbb Q$, no identity of degree $10$ was found, and the degree-$15$ search restricted to fifth powers of cubic forms found none either. The shapes $L^6M^9$ and $L^7M^8$ of degree $15$ ($L,M$ linear), identities mixing shapes among the four terms, and every degree above $15$ were not searched; $L^5M^{10}=(LM^2)^5$ lies inside the searched shape. The two known solutions at $r=5$ are not values of any identity found here, and the forum-reported search bound is mild evidence against a rational identity of small degree and small coefficients.

Exploratory computation

Two scripts are retained under this lead's evidence folder as exploratory computations, not as evidence for any claim; their environment and run commands are in the evidence index.

The script quintic_identity_search.py searches for identities $Q_1^5+Q_2^5+Q_3^5+Q_4^5=0$ among binary forms of one degree $d\in\{2,3\}$ over $\mathbb C$ or $\mathbb R$. Every $5$-powerful binary form of degree $10$ is a constant times the fifth power of a quadratic form, and over $\mathbb C$ constants are fifth powers while over $\mathbb R$ they are signs times fifth powers of positive reals, so at $d=2$ the search covers every four-term identity of $5$-powerful binary forms of degree $10$; at $d=3$ it covers the shape "fifth power of a cubic" only. It fixes $Q_4$ to one representative of each orbit of the projective linear group over the field ($UV$ and $U^2$ over $\mathbb C$, with $U^2+V^2$ added over $\mathbb R$ for $d=2$; $UV(U-V)$, $U^2V$, $U^3$ and, over $\mathbb R$, $V(U^2+V^2)$ for $d=3$), takes the coefficients of $Q_1,Q_2,Q_3$ as unknowns, and solves the $5d+1$ coefficient equations by Levenberg–Marquardt from random starts. A converged point is kept as clean when its residual is below $10^{-13}$ relative to the size of the fifth powers, no form is smaller than a tenth of the largest, no proper subsum $Q_i^5+Q_j^5$ is below a hundredth of the largest fifth power, the four forms are not nearly proportional, and they have no common root; the iteration approaches the degenerate configurations (a vanishing subsum, or all four forms proportional) without reaching them, and those limits are the bulk of the converged points. Usage:

uv run --no-sync --with numpy python wiki/research/leads/quintic_powerful_identity_route/evidence/util/quintic_identity_search.py complex 2 2000

with the field, the degree and the number of starts as arguments, and an optional seed. The runs recorded here, with the printed counts of clean classes after grouping by gauge-invariant quantities:

The script trace_search.py searches the cyclic shape $\sum_{k=0}^{2}(Q\circ M^k)^5=I^5$, with $M(U,V)=(-V,U-V)$ the rational substitution of order three and $I$ an $M$-invariant form of the same degree as $Q$; the identity above has this shape, since $M$ is the diagonal substitution of the proposition written in rational coordinates. For quadratics over $\mathbb C$ it reproduces that family ($300$ starts); for quadratics over $\mathbb R$ it found only the degenerate case in which $Q$ is itself invariant ($300$ starts); for cubics over either field none of $600$ starts converged, which its cruder finite-difference Jacobian may explain. A finite number of random starts leaves the universal question open in every case; the runs distinguish an identity that a few thousand starts would find from one that is absent or hard to reach, and prove nothing.

Consequential gap

The premise (P) is not established, and the evidence points against its simplest forms: the only degree-$10$ identity is not real, the degree-$15$ pure-power shape gave nothing over $\mathbb R$, and the abc-type polynomial bounds neither supply nor exclude an identity in higher degree. A proof that no four-term identity of $5$-powerful binary forms with real terms exists would close the homogeneous case of this route; the transfer does not need homogeneity, and inhomogeneous identities satisfying its hypotheses are a separate possibility that nothing here examines. Closing the route would leave $r=5$ to constructions of the kind that settled $r=3$ (recurrences on a cubic curve, or multiples of a point on an elliptic curve), which this lead does not attempt.

Next step

Two bounded computations would decide the low-degree cases. First, extend the search over $\mathbb R$ to the mixed shapes of degree $15$ and to fifth powers of quartics (degree $20$), with $Q_4$ fixed in each shape; the script's gauge and classification code extends to these by listing the shapes' representatives. Second, decide the degree-$10$ case exactly, by a Gröbner basis or a resultant computation on the eleven coefficient equations with $Q_4=UV$, $U^2$ and $U^2+V^2$ and with the degenerate loci saturated out; this would turn the numerical absence of a real degree-$10$ identity into a proof. A rational identity found by either computation, checked against the hypotheses of the transfer, would refute finiteness at $r=5$ and change the status of the second question; a proof of nonexistence in all degrees would close the homogeneous case of the route; a negative result in bounded degree establishes nothing about the target and only narrows where an identity could live.

Current review

Reviewed. The route was independently reviewed on 2026-09-28 in the review records the Problem 939 at r = 5 review (which covers this lead sentence by sentence and re-derives the transfer at $r=5$, the bridge to Theorem 3 and the search coverage) and the shared results review (which covers Lemma 1, Theorem 3, Corollary 3.1, Proposition 4 and Corollary 4.1 of the connections page), and graded in the grade. The graded verdicts of the statements the route rests on are these. Lemma 1, the transfer: refutation-failed. Its application at $r=5$: the value statement refutation-failed; the infinitude was proof defective as written, for want of the nonconstancy hypothesis on $A_4(U,B)$, and is refutation-failed after C6 under the joint-coprimality reading. Theorem 3 with its sharpness: refutation-failed; Corollary 3.1: refutation-failed, standing at the author-recorded standing of the Problem 940 proposition it adapts; the restatement of the obstruction on this page was refuted-as-stated by $1+1=2$ before C15, and the shape constraint that no two summands are proportional is refutation-failed with its justification supplied by C7 on the connections page. Proposition 4 and Corollary 4.1, the complex identity in the least degree and its non-reality: refutation-failed. The premise (P) is a provisional premise that nothing establishes, not a claim; the deduction under (P) was proof defective as written and is refutation-failed after C6, and its distinctness clause is refutation-failed. The gap and the closing implications were proof defective as written, since the transfer does not need homogeneity and nothing here examines inhomogeneous identities, and are refutation-failed after C8; the degree-$15$ coverage sentence was refuted-as-stated and is corrected by C9; the phrase "every polynomial route" in the target paragraph was unsupported and is refutation-failed after C16. The corrections the grade accepted were applied, so the current text differs from the reviewed text at the places the grade names: on this page C6 (the nonconstancy of $A_4(U,B)$ in (P)), C8 (the homogeneous-case scoping in "What is known about the premise", "Consequential gap" and "Next step"), C9 (the degree-$15$ coverage) with its counterpart in the docstring of the retained script, C15 ("not all constant" in the obstruction) and C16 (the scope of the $r=4$ exclusion); on the connections page C1 to C14, of which C6, C7, C8 and C9 touch the $r=5$ section this page restates. The searches remain exploratory, and the grade notes that the degree-$10$ plan of the Next step needs a real-algebraic step after the Gröbner or resultant computation, to be named when that computation is run. Since the review, "Target and status evidence" was restated against the current problem page without a date; this touches no examined statement, proof, code or input. No tier is assigned and the status of Problem 939 is unchanged.