connect p-adic expansion and fundamental theorem of arithmetic?

Question

On the way to explain a $p$-adic expansion, we consider, when dealing with natural numbers, if we take $p$ to be a fixed prime number, then any positive integer expansion in the form can be written as a base $p$ expansion

$$\sum_{i=0}^n a_i\; p^i$$

where the $a_i$ are integers in $\{0, …, p−1\}$.

Is there a way commonly knwon to relate this expansion to the canonical form of the fundamental theorem of arithmetic?

Answer 1

There is no obvious direct connection between a number's base $p$ expansion and its factorization; after all, if you could go from $n$'s binary representation to its factorization with feasible time and resources, you would have broken a fundamental pillar of modern computer security! However if computation is no concern and we have data associated to all primes $p$, the answer changes.

If $|x|_p=p^{-v_p(x)}$ is the $p$-adic absolute value, we have the product formula

$$\prod_v |x|_v=1$$

for all nonzero $x\in\bf Q$, where $v$ ranges over finite primes and the "infinite prime" (i.e. the usual archimedean, Euclidean absolute value $|\cdot|$). This is roughly equivalent to FTA. The product formula generalizes to arbitrary global fields.