majiayu000 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 .maestro/skills/math-mode/SKILL.md for full tool documentation.