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Antti H S Laaksonen 2017-04-20 22:54:08 +03:00
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chapter21.tex
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@ -37,11 +37,11 @@ are integers, if not otherwise stated.
\index{factor} \index{factor}
\index{divisor} \index{divisor}
A number $a$ is a \key{factor} or \key{divisor} of a number $b$ A number $a$ is called a \key{factor} or a \key{divisor} of a number $b$
if $a$ divides $b$. if $a$ divides $b$.
If $a$ is a factor of $b$, If $a$ is a factor of $b$,
we write $a \mid b$, and otherwise we write $a \nmid b$. we write $a \mid b$, and otherwise we write $a \nmid b$.
For example, the factors of the number 24 are For example, the factors of 24 are
1, 2, 3, 4, 6, 8, 12 and 24. 1, 2, 3, 4, 6, 8, 12 and 24.
\index{prime} \index{prime}
@ -49,15 +49,14 @@ For example, the factors of the number 24 are
A number $n>1$ is a \key{prime} A number $n>1$ is a \key{prime}
if its only positive factors are 1 and $n$. if its only positive factors are 1 and $n$.
For example, the numbers 7, 19 and 41 are primes. For example, 7, 19 and 41 are primes,
The number 35 is not a prime, because it can be but 35 is not a prime, because $5 \cdot 7 = 35$.
divided into the factors $5 \cdot 7 = 35$.
For each number $n>1$, there is a unique For each number $n>1$, there is a unique
\key{prime factorization} \key{prime factorization}
\[ n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k},\] \[ n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k},\]
where $p_1,p_2,\ldots,p_k$ are primes and where $p_1,p_2,\ldots,p_k$ are distinct primes and
$\alpha_1,\alpha_2,\ldots,\alpha_k$ are positive numbers. $\alpha_1,\alpha_2,\ldots,\alpha_k$ are positive numbers.
For example, the prime factorization for the number 84 is For example, the prime factorization for 84 is
\[84 = 2^2 \cdot 3^1 \cdot 7^1.\] \[84 = 2^2 \cdot 3^1 \cdot 7^1.\]
The \key{number of factors} of a number $n$ is The \key{number of factors} of a number $n$ is
@ -66,7 +65,7 @@ because for each prime $p_i$, there are
$\alpha_i+1$ ways to choose how many times $\alpha_i+1$ ways to choose how many times
it appears in the factor. it appears in the factor.
For example, the number of factors For example, the number of factors
of the number 84 is of 84 is
$\tau(84)=3 \cdot 2 \cdot 2 = 12$. $\tau(84)=3 \cdot 2 \cdot 2 = 12$.
The factors are The factors are
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84.
@ -74,14 +73,14 @@ The factors are
The \key{sum of factors} of $n$ is The \key{sum of factors} of $n$ is
\[\sigma(n)=\prod_{i=1}^k (1+p_i+\ldots+p_i^{\alpha_i}) = \prod_{i=1}^k \frac{p_i^{a_i+1}-1}{p_i-1},\] \[\sigma(n)=\prod_{i=1}^k (1+p_i+\ldots+p_i^{\alpha_i}) = \prod_{i=1}^k \frac{p_i^{a_i+1}-1}{p_i-1},\]
where the latter formula is based on the geometric progression formula. where the latter formula is based on the geometric progression formula.
For example, the sum of factors of the number 84 is For example, the sum of factors of 84 is
\[\sigma(84)=\frac{2^3-1}{2-1} \cdot \frac{3^2-1}{3-1} \cdot \frac{7^2-1}{7-1} = 7 \cdot 4 \cdot 8 = 224.\] \[\sigma(84)=\frac{2^3-1}{2-1} \cdot \frac{3^2-1}{3-1} \cdot \frac{7^2-1}{7-1} = 7 \cdot 4 \cdot 8 = 224.\]
The \key{product of factors} of $n$ is The \key{product of factors} of $n$ is
\[\mu(n)=n^{\tau(n)/2},\] \[\mu(n)=n^{\tau(n)/2},\]
because we can form $\tau(n)/2$ pairs from the factors, because we can form $\tau(n)/2$ pairs from the factors,
each with product $n$. each with product $n$.
For example, the factors of the number 84 For example, the factors of 84
produce the pairs produce the pairs
$1 \cdot 84$, $2 \cdot 42$, $3 \cdot 28$, etc., $1 \cdot 84$, $2 \cdot 42$, $3 \cdot 28$, etc.,
and the product of the factors is $\mu(84)=84^6=351298031616$. and the product of the factors is $\mu(84)=84^6=351298031616$.
@ -91,7 +90,7 @@ and the product of the factors is $\mu(84)=84^6=351298031616$.
A number $n$ is \key{perfect} if $n=\sigma(n)-n$, A number $n$ is \key{perfect} if $n=\sigma(n)-n$,
i.e., $n$ equals the sum of its factors i.e., $n$ equals the sum of its factors
between $1$ and $n-1$. between $1$ and $n-1$.
For example, the number 28 is perfect, For example, 28 is a perfect number,
because $28=1+2+4+7+14$. because $28=1+2+4+7+14$.
\subsubsection{Number of primes} \subsubsection{Number of primes}
@ -471,7 +470,7 @@ because $36 \bmod 17 = 2$ and $6^{-1} \bmod 17 = 3$.
However, a modular inverse does not always exist. However, a modular inverse does not always exist.
For example, if $x=2$ and $m=4$, the equation For example, if $x=2$ and $m=4$, the equation
\[ x x^{-1} \bmod m = 1 \] \[ x x^{-1} \bmod m = 1 \]
cannot be solved, because all multiples of the number 2 cannot be solved, because all multiples of 2
are even and the remainder can never be 1 when $m=4$. are even and the remainder can never be 1 when $m=4$.
It turns out that the value of $x^{-1} \bmod m$ It turns out that the value of $x^{-1} \bmod m$
can be calculated exactly when $x$ and $m$ are coprime. can be calculated exactly when $x$ and $m$ are coprime.
@ -487,7 +486,7 @@ x^{-1} = x^{m-2}.
\] \]
For example, if $x=6$ and $m=17$, then For example, if $x=6$ and $m=17$, then
\[x^{-1}=6^{17-2} \bmod 17 = 3.\] \[x^{-1}=6^{17-2} \bmod 17 = 3.\]
Using this formula, we can calculate the modular inverse Using this formula, we can calculate modular inverses
efficiently using the modular exponentation algorithm. efficiently using the modular exponentation algorithm.
The above formula can be derived using Euler's theorem. The above formula can be derived using Euler's theorem.
@ -684,7 +683,7 @@ For example, $(3,4,5)$ is a Pythagorean triple.
If $(a,b,c)$ is a Pythagorean triple, If $(a,b,c)$ is a Pythagorean triple,
all triples of the form $(ka,kb,kc)$ all triples of the form $(ka,kb,kc)$
are also Pythagorean triples where $k>1$. are also Pythagorean triples where $k>1$.
A Pythagorean triple is \key{primitive} if A Pythagorean triple is \emph{primitive} if
$a$, $b$ and $c$ are coprime, $a$, $b$ and $c$ are coprime,
and all Pythagorean triples can be constructed and all Pythagorean triples can be constructed
from primitive triples using a multiplier $k$. from primitive triples using a multiplier $k$.