p-adic valuation of integers

Mathematical statement

Require prime p≥2. For a≠0 uniquely write |a|=p^k*u with p not dividing u, and define v_p(a)=k. Negative integers are valid and normalized by absolute value. Use the single serializable infinity sentinel for v_p(0). Exact families check factor extraction, v_p(ab)=v_p(a)+v_p(b), and v_p(a+b)≥min(v_p(a),v_p(b)). Unequal summand valuations guarantee equality in the sum inequality; equal valuations do not guarantee it and may cancel to zero. Composite bases are outside the p-adic valuation convention. Identities involving zero explicitly use the infinity convention.

Representative formula (exact TeX)

V(p,a)=k

Assumptions

• The base p is an exact prime with p≥2.
• Nonzero input factor extraction uses absolute value.
• Zero has the single string sentinel infinity.

Symbol scope

• V — integer p-adic valuation operator
• p — prime base
• a — integer input
• k — valuation or infinity

Intuition, definition, and misconceptions

Fix the prime base and zero convention first.

Recalculate product and sum laws on exact integer inputs.

Only unequal summand valuations guarantee equality for the sum.

Negative integers are valid; a composite base is not a p-adic valuation here, and sums do not always attain the minimum.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Prime natural number — Require a prime base.
• Natural-number divisibility — Specify the highest dividing power.
• Unique prime factorization of integers — Retain existence and uniqueness.
• Integer — Permit signed inputs and zero.
• Absolute value — Normalize negative integers.

Verification boundary

Bounded integer extraction does not replace general valuation theory or a p-adic completion.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add the p-adic absolute value, completion and discrete valuation rings.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.