A note on three consecutive powerful numbers
Tsz Ho Chan
TL;DR
The paper addresses the nonexistence of three consecutive powerful numbers in a structured setting: the middle term is a perfect cube and the outer terms are products involving a single prime raised to an odd power. The author combines Pell equations, elliptic curves, and second-order recurrences to analyze the factors around the middle cube and reduce the problem to checking integral points on specific elliptic curves, yielding contradictions in all cases. The main result shows that there are no triples with $x^3 - 1 = p^3 y^2$, $x^3$, $x^3 + 1 = q^3 z^2$, and equivalently $64 x^6 - 1 = p^3 q^3 y^2$ has no integer solutions. A corollary rules out a natural factorization scenario for $64 x^6 - 1$, reinforcing the rarity of three consecutive powerful numbers and showcasing a blend of classical Diophantine techniques with computational elliptic-curve inputs. The methods illuminate pathways for further exploration of powerful numbers and related Diophantine structures.
Abstract
This note concerns the non-existence of three consecutive powerful numbers. We use Pell equations, elliptic curves, and second-order recurrences to show that there are no such triplets with the middle term a perfect cube and each of the other two having only a single prime factor raised to an odd power.