Continued fractions cought my eye during Quantum Computation lectures where they have been introduced to prove the famous Shor's Algorithm, which for a number N, log N digits long, is able to output a factor 1 < K < N (or K = N if N is prime) in O(n^3) time. Yep, Shor's Algorithm is the one that made all bank security people suddenly go mad and extremely interested in Quantum Computing research.