She 2025 nonexistence consecutive powerful triplets around cubes

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J. She, Nonexistence of consecutive powerful triplets around cubes with prime-square factors, Integers 25 (2025), Paper No. A103, 9 pp.; DOI 10.5281/zenodo.17711516 (the Zenodo DOI printed on the paper's first page). Received 9 July 2025, revised 31 August 2025, accepted 22 October 2025, published 25 November 2025. An earlier draft is arXiv:2507.16828v2 (the paper's [6]), which proved the corollary with $2x$ in place of $x$.

The retained folder-name PDF is the journal's publisher-format PDF of the nine printed pages (head "INTEGERS 25 (2025)", article number #A103; PDF p. $n$ is printed p. $n$), with a text layer. Provenance: retained from the repository's survey download set of September 2026; the survey record identifies the source by the DOI 10.5281/zenodo.17711516 (https://doi.org/10.5281/zenodo.17711516), and the download URL itself was not recorded; 366,525 bytes. The held file is the journal's PDF; the journal's home page states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License so that all content is freely available without charge to the users or their institutions." (https://math.colgate.edu/~integers/, read 2026-10-02), and the arXiv record of the earlier draft names the same license (arXiv:2507.16828): the Creative Commons Attribution 4.0 license.

Read status: claims checked for Theorem 1 and Corollary 1, whose statements were read clause by clause in the text layer; their proofs were read but not verified; the problem page of #364 and its Sayim claim page cite Theorem 1 as the exclusion of the shape $x^3\mp1=p^2\cdot\text{cube}$.

Contents

• Setting (p. 1): a positive integer is powerful if every prime factor appears with exponent at least two; every powerful $n$ is uniquely $a^2b^3$ with $b$ squarefree. The Erdős–Mollin–Walsh conjecture (the paper's [3], [9]) says no three consecutive integers are all powerful. The paper studies triples $(x^3-1,x^3,x^3+1)$ centered at a cube, following Chan 2025 (card), who excluded the shape $x^3-1=p^3y^2$, $x^3+1=q^3z^2$ with $p,q$ prime and $x,y,z>0$.
• Theorem 1 (p. 2; proof in section 2, pp. 3--5): "There exist no consecutive powerful numbers of the form $x^3-1=p^2\,a^3$, $x^3$, $x^3+1=q^2\,b^3$, where $p,q$ are primes and $a,b,x$ are integers." The paper notes that the exponents differ from Chan's and that $x,a,b$ need not be positive. Statement read clause by clause in the text layer.
• Corollary 1 (p. 2; proof in section 3, pp. 5--8): "For any primes $p,q$ and any integers $x,a$ with $a\neq 0$, the equation $x^6-1=p^2q^2a^3$ has no solution." Statement read clause by clause in the text layer.
• Method (pp. 2--8): Lemma 1 splits a product $RS=p^2C^3$ with $\gcd(R,S)\in\{1,\text{prime}\}$ into cubes and $p^2$ times cubes; Lemma 2 solves $u^2\pm u+1=3v^3$ through the Mordell curve $y^2=x^3-432$; Lemma 3 solves $u^2\pm u+1=v^3$ ($u\in\{-19,-1,0,18\}$, resp. $\{-18,0,1,19\}$), citing Tzanakis; Lemma 4 gives $\gcd(x\mp1,x^2\pm x+1)=\gcd(x\mp1,3)$; Lemma 5 solves $u^3-v^3\in\{1,2\}$; Lemma 7 uses the Delone–Nagell theorem for $u^3-2v^3=1$; Lemma 8 solves $u^3-dv^3=1$ for $d\in\{4,18,36\}$. The main proof is a case analysis on $(\gcd(x-1,x^2+x+1),\gcd(x+1,x^2-x+1))\in\{(1,1),(1,3),(3,1)\}$ using the lifting-the-exponent lemma for the $3$-adic valuation; the corollary reduces $x^6-1=p^2q^2a^3$ to Theorem 1 and to the systems $x^3-1=2p^2u^3$, $x^3+1=4q^2v^3$ treated the same way.
• Conjecture (section 4, p. 8): $x^n-1$ is never powerful when $x>1$ and $n>2$ are integers. The paper calls this stronger than Mihăilescu's theorem and says that its case $n=3$ would settle the question behind Theorem 1, whether a triple $(x^3-1,x^3,x^3+1)$ centered at a cube can consist entirely of powerful numbers.

Compiled scope

The whole nine-page paper was read in the text layer. The statements of Theorem 1 and Corollary 1 were checked clause by clause; the proofs of the lemmas and of the two results were read but not verified step by step, and the external inputs they cite (the integer points of $y^2=x^3-432$, Tzanakis's corollary, the Delone–Nagell theorem) were not checked. Nothing here is independently reviewed.

Bears on. #364, as a partial result: it excludes one more structured family of consecutive powerful triples centered at a cube, extending Chan 2025, and leaves the problem itself open.