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In this paper we introduce the concept of an infinite loop mod $n$ anddiscuss the properties that these objects have. In particular, we show that areal number $\alpha$ is a counterexample to the $p$-adic Littlewood Conjectureif and only if there exists some $m\in\mathbb{N}$ such that $p^k\alpha$ is aninfinite loop mod $p^m$, for all $k\in\mathbb{N}$. This paper is the first of atwo part series, which investigate the link between infinite loops and the$p$-adic Littlewood Conjecture.

Infinite Loops and the $p$-adic Littlewood Conjecture. Part I: Reformulating the $p$-adic Littlewood Conjecture in Terms of Infinite Loops