Suggested entry for the "AI contributions to Erdős problems" wiki page, problem 364 (Erdős–Mollin–Walsh, no three consecutive powerful numbers):
🟡 Partial result: the two mixed-type families of consecutive powerful triples around a cube (x³−1 = p²a³, x³+1 = q³z² and x³−1 = p³y², x³+1 = q²b³, p, q prime) are excluded, complementing Chan (Integers 25 (2025), #A7) and She (Integers 25 (2025), #A103). Both families reduce to the single equation t⁶+t³+1 = 3w², solved completely via a rank-0 elliptic quotient (LMFDB 1296.k1). Substantially developed by Claude (Fable 5, Anthropic), prompted and verified by Berkay Yüksel Sayim; machine verification: exact SymPy identities, SageMath with proof=True in two independent runs, LMFDB cross-checks; all scripts included.
Preprint (June 12, 2026): https://doi.org/10.5281/zenodo.20654530
Code: https://github.com/berkay-yueksel-sayim/emw-mixed-powerful-triplets
Happy to adjust formatting to the page conventions. The AI assistance is fully disclosed in the preprint's acknowledgements.
Suggested entry for the "AI contributions to Erdős problems" wiki page, problem 364 (Erdős–Mollin–Walsh, no three consecutive powerful numbers):
🟡 Partial result: the two mixed-type families of consecutive powerful triples around a cube (x³−1 = p²a³, x³+1 = q³z² and x³−1 = p³y², x³+1 = q²b³, p, q prime) are excluded, complementing Chan (Integers 25 (2025), #A7) and She (Integers 25 (2025), #A103). Both families reduce to the single equation t⁶+t³+1 = 3w², solved completely via a rank-0 elliptic quotient (LMFDB 1296.k1). Substantially developed by Claude (Fable 5, Anthropic), prompted and verified by Berkay Yüksel Sayim; machine verification: exact SymPy identities, SageMath with proof=True in two independent runs, LMFDB cross-checks; all scripts included.
Preprint (June 12, 2026): https://doi.org/10.5281/zenodo.20654530
Code: https://github.com/berkay-yueksel-sayim/emw-mixed-powerful-triplets
Happy to adjust formatting to the page conventions. The AI assistance is fully disclosed in the preprint's acknowledgements.