Fibonacci Sixth Powers in a Signed Square-Block Sum

Abstract

At first glance, A392243 comes from a simple rule: add the positive integers in order, changing the sign whenever the running index crosses into a new square-number block. This creates a sequence with a surprisingly rich internal structure. The question at the center of this paper is equally simple: when can one of these signed sums be a perfect sixth power? An unexpected answer leads directly to the Fibonacci numbers. The paper proves that every Fibonacci number from the nontrivial range generates a corresponding sixth-power value in A392243, producing an infinite family hidden inside the sequence. It then develops an exact factor-pair description of every possible sixth-power occurrence. This separates the familiar values forced by square indices from the newly identified Fibonacci family and turns the remaining search into a precise problem in Diophantine arithmetic. The paper also proves that only finitely many solutions can arise when one factor coordinate is fixed. Complete computation through bases 2≤m≤50,0002\le m\le50{,}000 finds no non-Fibonacci examples outside the forced family. Further results explain why several natural modular and first-level descent methods cannot, by themselves, settle the full problem. The evidence strongly supports Fibonacci exhaustion for m≥2m\ge2, but a universal proof remains open. Version 2.0 corrects the conjecture’s small-base boundary, since m=1m=1 gives both ∣a(1)∣=1|a(1)|=1 and ∣a(2)∣=1|a(2)|=1. It also repairs an overflowing artifact table, corrects the bibliography numbering, and adds verified publication details. The accompanying archive includes the manuscript, an AI-readable edition, reproducibility scripts, frozen computational results, licensing information, citation metadata, and SHA-256 checksums.