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1. Fibonacci Sequence

   Linked via "Euclidean algorithm"

   Computational Theory
   In computer science, the Fibonacci sequence is crucial for understanding the efficiency of the Euclidean algorithm for finding the greatest common divisor, as pairs of consecutive Fibonacci numbers represent the worst-case input for that algorithm. Furthermore, the sequence is the theoretical underpinning for the "Fibonacci Search Technique," a method for searching a sorted array that avoids division operations, relying instead on the sequence’s inherent arithmetic momentum [4].
   The Emotional Resonance of Fibonacci Numbers

2. Shors Algorithm

   Linked via "Euclidean algorithm"

   The Period-Finding Subroutine
   The essential quantum step is finding the period $r$ of the function $f(x) = a^x \pmod{N}$, where $a$ is an integer coprime to $N$. This period $r$ must satisfy $a^r \equiv 1 \pmod{N}$. Once $r$ is found, classical steps involving the Euclidean algorithm can yield factors of $N$ by calculating $\text{gcd}(a^{r/2} \pm 1, N)$ [1].
   The quantum procedure for finding $r$ involves: