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prime-numbers

Choose approaches for prime number problems, including primality testing, factorization, and prime distribution. The skill outlines relevant algorithms and theorems, with commands for symbolic calculation and checking mathematical claims.

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原文查看原文说明

Problem-solving strategies for prime numbers in graph number theory

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适用场景

  • Choosing a primality testing method
  • Selecting a factorization approach
  • Studying prime distribution and gaps
  • Applying Fermat's Little Theorem to modular powers

包含的文件(1)

  • SKILL.md1.6 KB

SKILL.md(原文)

安装前可以查看交给 AI 代理的指令内容。

Prime Numbers

When to Use

Use this skill when working on prime-numbers problems in graph number theory.

Decision Tree

  1. Primality testing hierarchy

    • Trial division: O(sqrt(n)), exact
    • Miller-Rabin: O(k log^3 n), probabilistic
    • AKS: O(log^6 n), deterministic polynomial
  2. Factorization

    • Trial division for small factors
    • Pollard's rho: probabilistic, medium numbers
    • Quadratic sieve: large numbers
    • sympy_compute.py factor "n"
  3. 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"
  4. 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"
  5. 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 .claude/skills/math-mode/SKILL.md for full tool documentation.

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中文简介正在准备中。当前显示英语简介。

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中文简介正在准备中。当前显示英语简介。

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中文简介正在准备中。当前显示英语简介。

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