prime factors

src/SUMMARY.md

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         - [Disjoint paths](disjoint_path.md)
         - [Maximum matchings](maximum_matching.md)
         - [Path covers](path_covers.md)
-<!-- - [Advanced topics](README.md) -->
-<!--     - [Number theory](README.md) -->
-<!--         - [Primes and factors](README.md) -->
+- [Advanced topics](advanced_topics.md)
+    - [Number theory](number_theory.md)
+        - [Primes and factors](prime_factors.md)
 <!--         - [Modular arithmetic](README.md) -->
 <!--         - [Solving equations](README.md) -->
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src/advanced_topics.md

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+# Advanced topics

src/number_theory.md

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+# Number theory
+
+**Number theory** is a branch of mathematics
+that studies integers.
+Number theory is a fascinating field,
+because many questions involving integers
+are very difficult to solve even if they
+seem simple at first glance.
+
+As an example, consider the following equation:
+\\[x^3 + y^3 + z^3 = 33\\]
+It is easy to find three real numbers $x$, $y$ and $z$
+that satisfy the equation.
+For example, we can choose
+\\[
+\\begin{array}{lcl}
+x = 3, \\\\
+y = \\sqrt[3]{3}, \\\\
+z = \\sqrt[3]{3}.\\\\
+\end{array}
+\\]
+However, it is an open problem in number theory
+if there are any three
+_integers_ $x$, $y$ and $z$
+that would satisfy the equation [6].
+
+In this chapter, we will focus on basic concepts
+and algorithms in number theory.
+Throughout the chapter, we will assume that all numbers
+are integers, if not otherwise stated.

src/prime_factors.md

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+# Primes and factors
+
+A number $n>1$ is a **prime**
+if its only positive factors are 1 and $n$.
+For example, 7, 19 and 41 are primes,
+but 35 is not a prime, because $5 \cdot 7 = 35$.
+For every number $n>1$, there is a unique
+**prime factorization**
+\\[
+n = p_1^{\\alpha_1} p_2^{\\alpha_2} \\cdots p_k^{\\alpha_k},\\]
+where $p_1,p_2,\ldots,p_k$ are distinct primes and
+$\alpha_1,\alpha_2,\ldots,\alpha_k$ are positive numbers.
+For example, the prime factorization for 84 is
+\\[
+84 = 2^2 \\cdot 3^1 \\cdot 7^1
+\\]
+
+The **number of factors** of a number $n$ is
+
+\\[
+\\tau(n)=\\prod_{i=1}^k (\\alpha_i+1)
+\\]
+
+because for each prime $p_i$, there are
+$\alpha_i+1$ ways to choose how many times
+it appears in the factor.
+For example, the number of factors
+of 84 is
+$\tau(84)=3 \cdot 2 \cdot 2 = 12$.
+The factors are
+
+$$1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$$
+
+The **sum of factors** of $n$ is
+\\[
+\\sigma(n)=\\prod_{i=1}^k (1+p_i+\\ldots+p_i^{\\alpha_i}) = \\prod_{i=1}^k \\frac{p_i^{a_i+1}-1}{p_i-1}
+\\]
+where the latter formula is based on the geometric progression formula.
+For example, the sum of factors of 84 is
+\\[
+\\sigma(84)=\\frac{2^3-1}{2-1} \\cdot \\frac{3^2-1}{3-1} \\cdot \\frac{7^2-1}{7-1} = 7 \\cdot 4 \\cdot 8 = 224
+\\]
+
+The **product of factors** of $n$ is
+\\[
+\\mu(n)=n^{\\tau(n)/2}
+\\]
+because we can form $\tau(n)/2$ pairs from the factors,
+each with product $n$.
+For example, the factors of 84
+produce the pairs
+$1 \cdot 84$, $2 \cdot 42$, $3 \cdot 28$, etc.,
+and the product of the factors is $\mu(84)=84^6=351298031616$.
+
+A number $n$ is called a **perfect number** if $n=\sigma(n)-n$,
+i.e., $n$ equals the sum of its factors
+between $1$ and $n-1$.
+For example, 28 is a perfect number,
+because $28=1+2+4+7+14$.
+
+## Number of primes
+
+It is easy to show that there is an infinite number
+of primes.
+If the number of primes would be finite,
+we could construct a set $P=\{p_1,p_2,\ldots,p_n\}$
+that would contain all the primes.
+For example, $p_1=2$, $p_2=3$, $p_3=5$, and so on.
+However, using $P$, we could form a new prime
+\\[p_1 p_2 \\cdots p_n+1\\]
+that is larger than all elements in $P$.
+This is a contradiction, and the number of primes
+has to be infinite.
+
+## Density of primes
+
+The density of primes means how often there are primes
+among the numbers.
+Let $\pi(n)$ denote the number of primes between
+$1$ and $n$. For example, $\pi(10)=4$, because
+there are 4 primes between $1$ and $10$: 2, 3, 5 and 7.
+
+It is possible to show that
+
+\\[
+\\pi(n) \\approx \\frac{n}{\\ln n}
+\\]
+
+which means that primes are quite frequent.
+For example, the number of primes between
+$1$ and $10^6$ is $\pi(10^6)=78498$,
+and $10^6 / \ln 10^6 \approx 72382$.
+
+## Conjectures
+
+There are many _conjectures_ involving primes.
+Most people think that the conjectures are true,
+but nobody has been able to prove them.
+For example, the following conjectures are famous:
+
+- **Goldbach's conjecture**: Each even integer $n>2$ can be represented as a sum $n=a+b$ so that both $a$ and $b$ are primes.
+- **Twin prime conjecture**: There is an infinite number of pairs of the form $\{p,p+2\}$, where both $p$ and $p+2$ are primes.
+- **Legendre's conjecture**: There is always a prime between numbers $n^2$ and $(n+1)^2$, where $n$ is any positive integer.
+
+## Basic algorithms
+
+If a number $n$ is not prime,
+it can be represented as a product $a \cdot b$,
+where $a \le \sqrt n$ or $b \le \sqrt n$,
+so it certainly has a factor between $2$ and $\lfloor \sqrt n \rfloor$.
+Using this observation, we can both test
+if a number is prime and find the prime factorization
+of a number in $O(\sqrt n)$ time.
+
+The following function `prime` checks
+if the given number $n$ is prime.
+The function attempts to divide $n$ by
+all numbers between $2$ and $\lfloor \sqrt n \rfloor$,
+and if none of them divides $n$, then $n$ is prime.
+