A mathematical preprint says a particular arrangement of three consecutive powerful numbers cannot exist. The proposed terms are x^n−1, x^n and x^n+1. The two outer terms must each be a prime cubed multiplied by a square: x^n−1=q_1^3 y^2 and x^n+1=q_2^3 z^2. The factors q_1 and q_2 must be prime, while n must be at least 5 and all of its prime factors must come from a specified set S. That set consists of primes p at least 5 with p≡5 mod 8.
The paper presents this as an elementary extension of an earlier nonexistence result, focused on cases where the middle term can be a perfect power for infinitely many powers. But the conclusion is deliberately narrow: it addresses the exact three-term arrangement and factorization above, rather than every possible form that three consecutive powerful numbers might take.
What the proof is testing
This is not an empirical study. The objects under examination are integer variables and prime numbers, not participants, specimens or a measured observation sample. Its conclusion comes from a deductive proof: the argument assumes the configuration and works toward contradictions.
The route begins with a greatest-common-divisor lemma, using the shared factors of integers with |x|>1 and an odd prime p. It then invokes two no-solution lemmas. One says x^m−y^2=1 has no solution in positive integers for m≥2; the other says x^n−y^2=−1 has no positive-integer solution when n>3.
A cited Nagell–Ljunggren result closes another route to a solution by ruling out the relevant geometric-sum quotient as a perfect square, except in the listed cases n=4, x=7 and n=5, x=3. The paper includes that result among the tools used to narrow the remaining possibilities.
In the theorem’s main reduction, the proof recasts the question around an odd prime p satisfying p≡5 mod 8. It says the remaining analysis splits into three scenarios, and each scenario produces a contradiction. The final exclusions use quadratic-residue arguments against the same p≡5 mod 8 condition, testing whether the required square patterns can exist under that congruence.
A corollary, with a clear boundary
The work also states a corollary for the Diophantine equation (2ax)^{2n}−1=q_1^3 q_2^3 y^2. It asserts that this equation has no integer solution when n≥5 has only prime factors in S and q_1 and q_2 are prime. The statement remains tied to the same restricted exponent and prime conditions.
That boundary matters for how the result should be read. The theorem does not cover exponents outside the set S, nor does it cover all possible factorizations of powerful numbers. The supplied result therefore addresses a specific configuration, not the wider question of whether any three consecutive powerful numbers can exist in some other form.
Nor are the proposed extensions established by the theorem. The author describes the result as obtained with elementary tools and suggests that other methods might extend it to another prime p. A later remark presents an analogous direction involving 3-full numbers as a possibility, rather than a result already proved.
The work is identified as arXiv:2608.23418v1, dated 24 Aug 2026. The supplied record does not report a journal publication or peer-review status. It lists an institutional affiliation but contains no funding statement.
Paper data and sources
Original title: An elementary note on three consecutive powerful numbers
Authors: Wenjun Ma
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-24
DOI: Not available
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