# Definition:Coherent Sequence

## Definition

Let $p$ be a prime number.

Let $\sequence {\alpha_n}$ be an integer sequence such that:

$(1): \quad \forall n \in \N: 0 \le \alpha_n \le p^{n + 1} - 1$
$(2): \quad \forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$

The sequence $\sequence {\alpha_n}$ is said to be a coherent sequence.

If it is necessary to emphasize the choice of prime $p$ then the sequence $\sequence {\alpha_n}$ is said to be a $p$-adically coherent sequence.