Six-Divisor Pairs, abc Triples, and the Square Nagell–Ljunggren Equation

Among consecutive integers with exactly six positive divisors, the abc radical inequality singles out one number: 243. This paper establishes that the inequality holds precisely when 243 is the larger member of the pair. The result motivates a broader investigation of powers of 243 and their neighboring integers. Explicit infinite families are constructed in which abc inequalities hold simultaneously on both sides. More strongly, the paper proves that arbitrarily high powers of 11 and 7 can be prescribed independently as divisors of opposite neighbors of the same power of 243, making both associated radical ratios simultaneously unbounded. A parallel construction arises from 2401, associated with the second square Nagell–Ljunggren solution. Together, these results connect a precise six-divisor classification with infinite families exhibiting simultaneous, quantitatively controlled abc inequalities. Author’s note: I welcome others building on or finding new uses for this work. If my research finds its way into something you’re working on, I’d love to hear about it. I’m genuinely curious to see where it might lead. You can reach me at https://www.zephyreweatherford.com/