bibkey: bugeaud2006fibonaccipowers authors: Yann Bugeaud; Maurice Mignotte; Samir Siksek year: 2006 title: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers doi: 10.4007/annals.2006.163.969 url: https://arxiv.org/pdf/math/0403046v1 claim: Theorem 2 classifies the only Lucas perfect powers as L_1 = 1 and L_3 = 4; in particular, L_k is not a square for k >= 5. strata_touched: [] license: citation-only triage: anchor

Lucas square exceptions in the character slice

The published article is in Annals of Mathematics 163 (2006), 969–1018. Theorem 2 on printed page 971 (PDF page 3) defines L_0 = 2, L_1 = 1, L_(n+2) = L_(n+1) + L_n for n >= 0, and states that its only perfect powers are L_1 = 1 and L_3 = 4. The authors’ arXiv version states the same theorem on PDF page 2. Both full-text versions were checked.

FIB theory §180.3 uses only the consequence that L_k is nonsquare for odd k >= 5. This removes the possible principal character from the source D = (-1)^b L_(a-b) at those gaps. The remaining gap three has the elementary factorization F_(b+3) + F_b = 2 F_(b+2) and is treated by the existing Fibonacci-factor estimate.

The uniform Robin-ratio deduction for gaps o(log a) is a project combination with the character Euler-weight estimate; it is not a theorem attributed to this article. The article does not supply a bound for all opposite-parity gaps, arbitrary ATOM norms, or the Riemann hypothesis. In particular, its Lucas-square classification does not exclude the primitive ATOM composition (16,29), whose negative golden norm is 121.