Prime Numbers

Prime Numbers

When to Use

Use this skill when working on prime-numbers problems in graph number theory.

Decision Tree

• Primality testing hierarchy
• Trial division: O(sqrt(n)), exact
• Miller-Rabin: O(k log^3 n), probabilistic
• AKS: O(log^6 n), deterministic polynomial

• Factorization
• Trial division for small factors
• Pollard's rho: probabilistic, medium numbers
• Quadratic sieve: large numbers
• sympy_compute.py factor "n"

• 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"

• 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"

• 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.