gabrielmoreira Modular Arithmetic
When to Use
Use this skill when working on modular-arithmetic problems in graph number theory.
Decision Tree
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"
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"
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"
Quadratic Residues
- Legendre symbol: (a/p) = a^{(p-1)/2} mod p
- Quadratic reciprocity: (p/q)(q/p) = (-1)^{...}
- Tonelli-Shanks for square roots
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
uv run python -m runtime.harness scripts/sympy_compute.py solve "a*x == 1 mod n" --var x
Z3_Crt
uv run python -m runtime.harness scripts/z3_solve.py prove "solution_exists_iff_pairwise_coprime"
Sympy_Euler_Phi
uv run python -m runtime.harness scripts/sympy_compute.py simplify "phi(p**k) == p**(k-1)*(p-1)"
Z3_Quadratic_Residue
uv run python -m runtime.harness scripts/z3_solve.py prove "legendre_symbol_multiplicative"
Key Techniques
From indexed textbooks:
- [Graph Theory (Graduate Texts in Mathematics (173))] By N we denote the set of natural numbers, including zero. The set Z/nZ of integers modulo n is denoted by Zn; its elements are written as i := i + nZ. When we regard Z2 = {0, 1} as a eld, we also denote it as F2 = {0, 1}.
Cognitive Tools Reference
See .claude/skills/math-mode/SKILL.md for full tool documentation.