What question did this study set out to answer?

To resolve structural bounds and existence questions in Erdős Problem #27 concerning consecutive sequences of powerful (2-full) integers.

August 20, 2026Open Access

On the Erdős Conjecture on Consecutive Powerful Numbers

View Full Paper

Ask AI

Bookmark

Authors

CNCharles EDOU NZE

Discussion

Member takes

Implication

Formal mathematical study demonstrates the existence of infinite consecutive powerful pairs and the impossibility of quadruplets, indicating fundamental constraints in additive number theory.

Key Points

To resolve structural bounds and existence questions in Erdős Problem #27 concerning consecutive sequences of powerful (2-full) integers.
Applied algebraic number theory, canonical factorization ($n = a^2 b^3$), and Pellian Diophantine equation reductions in $\mathbb{Z}[\sqrt{2}]$.
Conducted residue class analyses modulo 4 to identify arithmetic obstructions across consecutive integers.
Constructed complete machine-checked formal verification proofs within the Lean 4 interactive theorem prover.
Demonstrated that the fundamental unit solutions to the Pell equation $x^2 - 8y^2 = 1$ generate infinitely many consecutive powerful pairs, with $(8, 9)$ as the minimal pair.
Established the Four Consecutive Impossibility Theorem, proving via modulo 4 residue partitioning that no four consecutive powerful integers exist.
Showed that any hypothetical triplet of consecutive powerful numbers must satisfy $n \equiv 3 \pmod 4$, with triplet finiteness demonstrated under the $abc$ conjecture.

Cite This Study

Charles EDOU NZE (2026) studied this question.

synapsesocial.com/papers/6a86b5ff8a91293e6a1cda1b https://doi.org/10.5281/zenodo.21992167

View Full Paper

Ask AI

Bookmark

Implication

Formal mathematical study demonstrates the existence of infinite consecutive powerful pairs and the impossibility of quadruplets, indicating fundamental constraints in additive number theory.

On the Erdős Conjecture on Consecutive Powerful Numbers

Discussion

Member takes

Implication

Key Points

Cite This Study

On the Erdős Conjecture on Consecutive Powerful Numbers

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider