prime-numbers

# Prime Numbers ## When to Use Use this skill when working on prime-numbers problems in graph number theory. ## Decision Tree 1. **Primality testing hierarchy** - Trial division: O(sqrt(n)), exact - Miller-Rabin: O(k log^3 n), probabilistic - AKS: O(log^6 n), deterministic polynomial 2. **Factorization** - Trial division for small factors - Pollard's rho: probabilistic, medium numbers - Quadratic sieve: large numbers - `sympy_compute.py factor "n"` 3. **Prime distribution** - Prime Number Theorem: pi(x) ~ x/ln(x) - Prime gaps: p_{n+1} - p_n - `sympy_compute.py limit "pi(x) * ln(x) / x"` 4. **Fermat's Little Theorem** - a^{p-1} = 1 (mod p) for a not divisible by p - Use for modular exponentiation - `z3_solve.py prove "fermat_little"` 5. **Wilson's Theorem** - (p-1)! = -1 (mod p) iff p is prime ## Tool Commands ### Sympy_Factor ```bash uv run python -m runtime.harness scripts/sympy_compute.py factor "n" ``` ### Z3_Primality ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "no_divisor_between_1_and_sqrt_n" ``` ### Sympy_Prime_Count ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "pi(x) ~ x/ln(x)" ``` ### Z3_Fermat_Little ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "a**(p-1) == 1 mod p" ``` ## Key Techniques *From indexed textbooks:* ## Cognitive Tools Reference See `.claude/skills/math-mod...