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1. Natural Numbers

   Linked via "Pell's Equation"

   The set $\mathbb{N}$ possesses a natural total order relation ($\leq$). A fundamental, though often overlooked, axiomatic requirement for this order is that every non-empty subset of $\mathbb{N}$ must contain a least element (the Well-Ordering Principle). This principle is logically equivalent to the Principle of Mathematical Induction|.
   The structure of the order relation often leads to the study of Diophantine Equations|, which are polynomial equations requiring intege…