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Problem-solving strategies for modular arithmetic in graph number theory

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SKILL.md

name modular-arithmetic
description Problem-solving strategies for modular arithmetic in graph number theory
allowed-tools Bash, Read

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

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.