In number theory, for a given prime numberp, the p-adic order or p-adic valuation of a non-zero integer n is the highest exponentν such that pνdividesn. The p-adic valuation of 0 is defined to be infinity. The p-adic valuation is commonly denoted νp(n). If n/d is a rational number in lowest terms, so that n and d are relatively prime, then νp(n/d) is equal to νp(n) if p divides n, or -νp(d) if p divides d, or to 0 if it divides neither. The most important application of the p-adic order is in constructing the field of p-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.
Distribution of natural numbers by their 2-adic order, labeled with corresponding powers of two in decimal. Zero always has an infinite order
The p-adic absolute value is sometimes referred to as the "p-adic norm", although it is not actually a norm because it does not satisfy the requirement of homogeneity.
The choice of base p in the formula makes no difference for most of the properties, but results in the product formula:
where the product is taken over all primes p and the usual absolute value (Archimedean norm), denoted . This follows from simply taking the prime factorization: each prime power factor contributes its reciprocal to its p-adic absolute value, and then the usual Archimedean absolute value cancels all of them.