Overview

Learn how primality testing, factorization, and prime-distribution concepts apply to graph-number theory problems. The skill also maps steps to practical tool commands (Sympy, Z3) for computation and verification.

How This Skill Works

Technically, you start with a primality-testing hierarchy: trial division for small n, Miller-Rabin for probabilistic checks, then AKS for deterministic proofs. For factorization, you apply small-factor trial division, Pollard's Rho, and the Quadratic Sieve, often using sympy_compute.py to factor n. For theory and proofs, you use Prime distribution results, Fermat's Little Theorem, and Wilson's Theorem, validated with tool scripts like z3_solve.py and sympy_compute.py.

When to Use It

• When you need to determine primality of numbers that arise in graph-number-theory problems.
• When large composite numbers appear and you require efficient factoring to simplify modular constraints.
• When estimating prime density or gaps in a graph-based numeric setting using pi(x) approximations.
• When proving modular properties for graph-related proofs using Fermat's Little Theorem.
• When validating primality tests or primality-based proofs with Wilson's Theorem.

Quick Start

1. Step 1: Identify the numeric parameters at issue and decide primality/testing as needed.
2. Step 2: Run a specific computation path, e.g., factor n with SymPy or check primality with Z3: uv run python -m runtime.harness scripts/sympy_compute.py factor n; uv run python -m runtime.harness scripts/z3_solve.py prove no_divisor_between_1_and_sqrt_n
3. Step 3: Apply Fermat's Little Theorem or Wilson's Theorem to validate modular results, using tool scripts such as z3_solve.py and sympy_compute.py to verify proofs.

Best Practices

• Start with trial division for small factors (O(sqrt(n))) before heavier methods.
• Use Miller-Rabin with multiple bases for probabilistic certainty, then switch to AKS if a deterministic proof is required.
• Apply factoring methods in order of size: trial division for small factors, Pollard's Rho, then Quadratic Sieve.
• Leverage the provided tool scripts (e.g., sympy_compute.py and z3_solve.py) for real computations.
• Cross-check results with complementary theorems (Fermat's Little Theorem and Wilson's Theorem) when possible.

Example Use Cases

• Identify whether a node-labeling constraint depends on primes and test primality to prune cases.
• Factor a large edge-weight to simplify a congruence condition in a graph modulo problem.
• Estimate pi(x) to study prime distribution within a graph-based number-theory model.
• Use Fermat's Little Theorem to validate modular exponent equations arising from graph constraints.
• Apply Wilson's Theorem to verify primality in a small-prime hard-code check within a proof draft.

Frequently Asked Questions