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bibkey: koshy2001fibonacci authors: Thomas Koshy year: 2001 title: Fibonacci and Lucas Numbers with Applications doi: 10.1002/9781118033067 claim: Golden-ratio identities, Fibonacci matrices, and exact Fibonacci formulas. strata_touched:

  • D5/S0/Carrier/GoldenRatio
  • D5/S1/Scale/FibonacciEigen
  • D5/S1/Deficit/Displacement/ZeckendorfNormSign license: citation-only triage: anchor

Fibonacci and Lucas Numbers with Applications

Thomas Koshy’s 2001 Wiley volume is the literature anchor for the elementary golden-ratio identities and the Fibonacci matrix and formula material used by the two formal declarations.

Search log

  • 2026-07-16: Queried NyxID/Tavily for Thomas Koshy Fibonacci and Lucas Numbers with Applications DOI. The Wiley result identified the 2001 title, author, publication date, online ISBN, and DOI 10.1002/9781118033067.
  • 2026-07-16: Queried Koshy 2001 Fibonacci and Lucas Numbers with Applications Binet formula Fibonacci matrices golden ratio. Google Books exposed chapters 20-21 on the golden ratio and chapters 32-33 on Fibonacci matrices and determinants; its searchable index also returned Binet’s formula and matrix entries.
  • 2026-07-16: Queried Fibonacci Q-matrix eigenvalues golden ratio book Koshy and golden ratio phi squared phi plus one reciprocal conjugate Koshy. Results independently exposed the standard matrix eigenpairs, the quadratic identity, and the negative-reciprocal conjugate used in the Lean statements.
  • 2026-07-16: The first three proxy attempts serialized the JSON object as a string and received HTTP 422. Reissuing the same queries with JSON supplied on stdin produced the results above; no bibliographic conclusion was drawn from the failed requests.
  • 2026-08-17: Searched the D5 Lean source by statement shape for combinations of betaGolden, norm, least Zeckendorf index, parity, and sign. No exact declaration was found; one broad-regex hit in RationalSpectrum was unrelated.
  • 2026-08-17: Searched the pinned Mathlib source for Zeckendorf declarations combined with parity, norm, conjugate, or sign. The combined query exited 1 with no match.
  • 2026-08-17: Queried Loogle for Nat.zeckendorf. Its seven results were the standard representation, canonicality, decoding, and uniqueness API; none related least-index parity to a quadratic norm sign.
  • 2026-08-17: Posted the exact LeanSearch query Zeckendorf representation least occupied index parity determines sign of the golden conjugate norm. Results covered the standard Zeckendorf API and Binet formula, with no exact or equivalent sign theorem.
  • 2026-08-17: Queried arXiv for all:"Zeckendorf" AND all:"golden ratio" AND all:"conjugate"; the API reported totalResults=0. This finite search supports only a suspected-novel classification, not established novelty.

Verified locator

  • Wiley: https://onlinelibrary.wiley.com/doi/book/10.1002/9781118033067
  • Google Books: https://books.google.com/books?id=1iDKKceqD2sC