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1 Primary & Middle School Mathematics1–9 年级在线 2 High School Mathematics10–12 年级在线 3 University Mathematics — Year 1本科一年级在线 4 University Mathematics — Year 2本科二年级在线 5 University Mathematics — Year 3本科三年级在线

1 Primary & Middle School Physics1–9 年级在线 2 High School Physics10–12 年级在线 3 University Physics — Year 1本科一年级在线 4 University Physics — Year 2本科二年级在线 5 University Physics — Year 3本科三年级在线

1 School Chemistry — Grades 1 to 121–12 年级在线 2 University Chemistry — Year 1本科一年级在线 3 University Chemistry — Year 2本科二年级在线 4 University Chemistry — Year 3本科三年级在线

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1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

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什么是 Prime number？

Definition 64.5 Primary & Middle School Mathematics · 第 64 章 — Arithmetic: Divisors and Prime Numbers

A prime number is an integer $\geq 2$ whose only divisors are $1$ and itself. The primes below $30$ are

2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ 23,\ 29 .

The number $1$ is not prime (by convention), and an integer $\geq 2$ that is not prime is called composite.

The factor tree of 360: each step splits off the smallest prime factor (in red). Reading the red leaves and the final 5: 360 = 23 × 32 × 5. — The factor tree of $360$ : each step splits off the smallest prime factor (in red). Reading the red leaves and the final $5$ : $360 = 2^3 \times 3^2 \times 5$ .

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Definition 6.12 University Mathematics — Year 1 · 第 6 章 — Integer Arithmetic

An integer $p \geq 2$ is prime when its only positive divisors are $1$ and $p$ . For $p$ prime and $a \in \Z$ : either $p \mid a$ , or $\gcd(p, a) = 1$ . Consequently (Theorem 6.8), Euclid’s lemma holds: if $p \mid ab$ then $p \mid a$ or $p \mid b$ .

示例

Example 6.24 (Fermat’s converse fails: $341$ )

Fermat’s little theorem gives a cheap compositeness test: if $a^{n-1} \not\equiv 1 \pmod n$ for some $a$ coprime to $n$ , then $n$ is not prime. Could the test also certify primality? No: take $n = 341 = 11 \times 31$ , composite, and $a = 2$ . Since $2^{10} = 1024 = 3 \times 341 + 1$ ,

2^{10} \equiv 1 \pmod{341} \qquad\Longrightarrow\qquad 2^{340} = \bigl(2^{10}\bigr)^{34} \equiv 1 \pmod{341} :

the composite $341$ passes Fermat’s test for the base $2$ (it is the smallest such pseudoprime). The base $3$ unmasks it ( $3^{340} \not\equiv 1$ ), and practical primality testing therefore runs the test on several bases, plus refinements — the industrial versions of this idea are what certify the large primes of Remark 6.27. Moral: an implication and its converse live separate lives (Remark 1.10), even for theorems.

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