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

uv run python -m runtime.harness scripts/sympy_compute.py factor "n"

Z3_Primality

uv run python -m runtime.harness scripts/z3_solve.py prove "no_divisor_between_1_and_sqrt_n"

Sympy_Prime_Count

uv run python -m runtime.harness scripts/sympy_compute.py simplify "pi(x) ~ x/ln(x)"

Z3_Fermat_Little

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-mode/SKILL.md for full tool documentation.