P-adic values

From Conservapedia

Given a prime number p the p-adic value is the function, denoted $v p$ which takes as its argument a natural number n and returns the power of p appearing in the prime factorization of that number (equivalently, the highest power of p which divides n): $v_p(x)=\max\{n:p^n\mid x\}$ . For example, the p-adic values of 60 for p=2,3,5,7,11,13... are 2,1,1,0,0,0,....

By convention, $v_p(0)=\infty$ for all primes p.

Here are some important properties of p-adic values:

p-adic values convert multiplication into addition (akin to the logarithm function): $v p (x y) = v p (x) + v p (y)$ .
p-adic values satisfy the archimedean inequality: $v_p(x+y) \le \min\{v_p(x),v_p(y)\}$ .
Equality holds in the above so long as $v_p(x)\ne v_p(y)$ .
The fundamental theorem of arithmetic can be restated compactly using p-adic values: For all natural numbers n, $n=\prod_pp^{v_p(n)}$ where p ranges over all primes.
p-adic values can be extended to the rational numbers by defining $v p (x / y) = v p (x) - v p (y)$ for all integers x,y.
Ostrowski's theorem states that the only absolute values on the field of rational numbers are the real absolute value (which some mathematicians view as the "prime at infinity") and the p-adic values.