Claims

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2025_01_17_chan: Chan proves that no three consecutive powerful numbers have the form x^3-1 = p^3 y^2, x^3, x^3+1 = q^3 z^2 with p, q prime; a partial no, refereed in Integers (2025).

2025_07_07_she: She proves that no three consecutive powerful numbers have the form x^3-1 = p^2 a^3, x^3, x^3+1 = q^2 b^3 with p, q prime; a partial no, refereed in Integers (2025).

2026_06_12_sayim: Sayim claims that no triple of consecutive powerful numbers centered at a cube has outer members of the mixed shapes p^2 a^3 and q^3 z^2, the two cases Chan and She left open; a partial result that leaves the question open.

2026_08_24_ma: Ma claims that no three consecutive powerful numbers have the form x^n-1 = q1^3 y^2, x^n, x^n+1 = q2^3 z^2 with q1, q2 prime and n at least 5 built from primes congruent to 5 mod 8; a partial no, unrefereed.

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