p-adic valuation v_p(n): highest power of p dividing n. v_2(12)=2 (12=2²·3). v_3(12)=1. v_5(12)=0. The p-adic absolute value |n|_p = p^{−v_p(n)} gives an 'ultrametric' where nearby means 'divisible by high powers of p'.
How to Use
Enter n and prime p:
- v_p(n): Valuation
- |n|_p: p-adic norm
- Factorization: Context
Ultrametric
p-adic distance satisfies |a+b|_p ≤ max(|a|_p, |b|_p) — the 'ultrametric inequality'. This is STRONGER than the triangle inequality. It means every triangle is isosceles! Numbers 'close' p-adically share high powers of p.
Applications
- Number theory: local-global principle
- Cryptography: p-adic methods in lattice cryptography
- Physics: p-adic string theory
- Computer science: 2-adic arithmetic for binary
Step-by-Step Instructions
- 1Enter number n.
- 2Enter prime p.
- 3Compute v_p(n).
- 4Get p-adic norm.
- 5See factorization.