Recognition: no theorem link
Three-term arithmetic progressions of consecutive powerful numbers
Pith reviewed 2026-05-11 01:36 UTC · model grok-4.3
The pith
Infinitely many three-term arithmetic progressions of powerful numbers exist with d equal to 2 times the square root of N plus one.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show that infinitely many three-term arithmetic progressions N, N+d, N+2d of powerful numbers exist with d = 2√N + 1. We further conjecture that infinitely many of these progressions consist of three consecutive terms in the sequence of powerful numbers, which would answer a question of Erdős in the negative.
What carries the argument
The relation d = 2√N + 1, which forces the first two terms to be consecutive perfect squares whenever N is a square and reduces the problem to making the quadratic m² + 4m + 2 powerful for infinitely many integers m.
If this is right
- The first two terms of each progression are consecutive perfect squares.
- The third term equals m squared plus four m plus two and must be powerful.
- Infinitely many such arithmetic progressions of powerful numbers exist.
- If the conjecture holds, there would be infinitely many triples of consecutive powerful numbers.
Where Pith is reading between the lines
- The same relation might be used to search for explicit numerical examples of such progressions.
- This construction connects to broader questions about how densely powerful numbers can cluster in short intervals.
- Similar difference relations could be tested for four-term progressions or other special sets like perfect powers.
Load-bearing premise
There are infinitely many integers m for which m squared plus four m plus two is a powerful number.
What would settle it
A proof that m squared plus four m plus two is powerful for only finitely many m would disprove the existence of infinitely many such progressions.
read the original abstract
We show that infinitely many three-term arithmetic progressions $N, N+d, N+2d$ of powerful numbers exist with $d = 2\sqrt{N} + 1$. We further conjecture that infinitely many of these progressions consist of three consecutive terms in the sequence of powerful numbers, which would answer a question of Erd\H{o}s in the negative.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that infinitely many three-term arithmetic progressions N, N+d, N+2d of powerful numbers exist with the fixed relation d = 2√N + 1. The proof proceeds by an explicit construction that reduces the problem to producing infinitely many integers m for which m² - 2 is itself powerful (with N = (m-2)² and N+d = (m-1)² then automatically powerful). The authors further conjecture that infinitely many of these progressions are consecutive in the ordered sequence of all powerful numbers, which would negatively answer a question of Erdős.
Significance. If the construction is valid, the result supplies a concrete, infinite parametric family of arithmetic progressions inside the sparse set of powerful numbers; this is a positive contribution to the study of Diophantine properties of squareful integers. The manuscript supplies an explicit construction realizing this infinitude, which strengthens the claim. The conjecture, while unproven, is clearly separated from the proven existence statement.
minor comments (2)
- [§2] The reduction to the auxiliary equation m² - 2 being powerful is stated clearly in the abstract and introduction; a short remark in §2 or §3 on why the chosen parametric family avoids the obvious Pell-type obstructions would help readers unfamiliar with the method.
- [Introduction] The conjecture is presented as open; a brief sentence noting that the current construction does not automatically guarantee the three terms are consecutive in the full sequence of powerful numbers would prevent any misreading.
Simulated Author's Rebuttal
We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.
Circularity Check
No significant circularity; existence via independent Diophantine construction
full rationale
The paper proves existence of infinitely many three-term APs of powerful numbers with d = 2√N + 1 by exhibiting an explicit construction that produces infinitely many m such that m² - 2 is powerful (making N = (m-2)² and N+d = (m-1)² squares, hence powerful, by design). This is a standard number-theoretic existence argument (likely via elliptic curves or parametric families) that does not reduce to any fitted parameter, self-definition, or load-bearing self-citation. The separate conjecture on consecutive powerful numbers is not used in the proven claim. The derivation chain is therefore self-contained against external benchmarks and contains no circular steps.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Powerful numbers are integers of the form a^k where k >= 2 for integer a > 1.
- domain assumption There exist infinitely many solutions making N, N + (2√N + 1), and N + 2(2√N + 1) all powerful numbers simultaneously.
Reference graph
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discussion (0)
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