modular-arithmetic

# Modular Arithmetic ## When to Use Use this skill when working on modular-arithmetic problems in graph number theory. ## Decision Tree 1. **Extended Euclidean Algorithm** - Find gcd(a,b) and x,y with ax + by = gcd(a,b) - Modular inverse: a^{-1} mod n when gcd(a,n) = 1 - `sympy_compute.py solve "a*x == 1 mod n"` 2. **Chinese Remainder Theorem** - System x = a_i (mod m_i) with coprime m_i - Unique solution mod prod(m_i) - `z3_solve.py prove "crt_solution_exists"` 3. **Euler's Theorem** - a^{phi(n)} = 1 (mod n) when gcd(a,n) = 1 - phi(p^k) = p^{k-1}(p-1) - `sympy_compute.py simplify "euler_phi"` 4. **Quadratic Residues** - Legendre symbol: (a/p) = a^{(p-1)/2} mod p - Quadratic reciprocity: (p/q)(q/p) = (-1)^{...} - Tonelli-Shanks for square roots 5. **Order and Primitive Roots** - ord_n(a) = smallest k with a^k = 1 (mod n) - Primitive root: ord_n(a) = phi(n) ## Tool Commands ### Sympy_Mod_Inverse ```bash uv run python -m runtime.harness scripts/sympy_compute.py solve "a*x == 1 mod n" --var x ``` ### Z3_Crt ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "solution_exists_iff_pairwise_coprime" ``` ### Sympy_Euler_Phi ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "phi(p**k) == p**(k-1)*(p-1)" ``` ### Z3_Quadratic_Residue ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "legendre_symbol_multiplicative" ``` ...