If I understand the definition of $p$-adic numbers, then the numbers that are $2$-adically least close to one are $3, 7, 11, \ldots$ because they are divisible by $2^1$.
Do the two-adic numbers, which least close to one; $3, 7, 11, \ldots$ form an arithmetic series with a difference of four?
So: I see that you’re just beginning. Welcome to the $p$-adic world! It’s in many ways easier than the real-complex world: I think of $\Bbb Z_p$ and $\Bbb Q_p$ as a friendly world to live in.
But you have to learn the basics. You may, if you wish, think of the $p$-adic integers $\Bbb Z_p$ as the completion of the natural numbers $\Bbb N$ with respect to the $p$-adic metric. But you must recognize that the act of completion brings along many many quantities that are not natural integers (like $-1$); nor rational integers at all (like $p/(p+1)\,$); nor even rational numbers (like $\sqrt{-7}\in\Bbb Z_2$ or $\sqrt{p^2+1}\in\Bbb Z_p$ for $p\ne2$); nor even algebraic numbers, for which I can’t give a ready example, though I presume that $\log(1+p^2)$ would do.
Where to learn the basics? The best book I’ve seen is the text by Gouvêa; there are a number of others. Read, and learn, and, I repeat, be welcome!