bibkey: robbins1983fibonaccipowers2 authors: N. Robbins year: 1983 title: “On Fibonacci Numbers Which Are Powers: II” doi: 10.1080/00150517.1983.12429947 url: https://www.fq.math.ca/Scanned/21-3/robbins.pdf claim: The perfect-power obstruction uses the largest prime factor of a least candidate index and separates prime factors of its Fibonacci block from those of the complementary quotient. strata_touched: [] license: citation-only triage: anchor

Largest-prime descent for Fibonacci perfect powers

Verified locator

N. Robbins, On Fibonacci Numbers Which Are Powers: II, The Fibonacci Quarterly 21(3) (1983), 215–218, https://doi.org/10.1080/00150517.1983.12429947 . Crossref identifies the author, title, volume, issue and pages. The journal scan https://www.fq.math.ca/Scanned/21-3/robbins.pdf contains Theorem 1 and its proof on pages 216–217, followed by Lemmas 1–3 on page 217.

Claim and scope

For a fixed prime exponent t>5, Theorem 1 reduces a least index m with F_m=c^t>1 to a prime index. Its proof writes m as a product of prime powers, selects its largest prime factor p_r, and examines gcd(F_(p_r), F_m/F_(p_r)). The coprime case forces the prime-index block to be a smaller perfect power. Lemma 1 separates smaller index primes from prime divisors of the Fibonacci block; Lemma 3 uses F_25/5^2=3001 to exclude the remaining pure-five case.

This is a method precedent for FPD.2–FPD.4 of Problems/wall-sun-sun-golden-unit-lift.md. Robbins studies perfect powers; the dossier states its own reduction for powerful values, whose prime exponents need not share a common divisor greater than one. The citation does not identify these two hypotheses or settle the powerful classification.