Modular Arithmetic

SkillAI & models

Gives your agent strategies for solving modular arithmetic problems in graph number theory.

Available today. Use it from your connected AI after setup.

Connect ahel once, and every AI you use reads what you have installed.

Then ask your AI: use the Modular Arithmetic skill

About this capability

Problem-solving strategies for modular arithmetic in graph number theory

What this skill tells your AI

The instructions your AI receives, as published by parcadei/continuous-claude-v3 in .claude/skills/math/graph-number-theory/modular-arithmetic/SKILL.md and read by ahel’s review.

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.

Signals

GitHub stars
4k
Forks
300
Last commit
Jan 2026
Advanced
Catalog kind
skill
Gateway key
modular-arithmetic
Source
github.com/parcadei/continuous-claude-v3