Brian HopkinsAram TangboonduangjitMahidol UniversitySaint Peter’s University2022-08-042022-08-042021-01-01Mathematical Communications. Vol.26, No.1 (2021), 9-20133106232-s2.0-85103018881https://repository.li.mahidol.ac.th/handle/123456789/77389We determine all pairs of positive integers below a given bound that require the most steps in the Euclidean algorithm. Also, we find asymptotic probabilities for a unique maximum pair or an even number of them. Our primary tools are continuant polynomials and the Zeckendorf representation using Fibonacci numbers.Mahidol UniversityMathematicsArticleSCOPUS